AN  ANALYSIS  OF  PUPILS’  MISTAKES 
IN  GEOMETRY 


By 

GRACE  ERMINIE  MADDEN 

A.  B.  University  of  Illinois,  1917 


THESIS 


SUBMITTED  IN  PARTIAL  FULFILLMENT  OF  THE  REQUIREMENTS 
FOR  THE  DEGREE  OF  MASTER  OF  ARTS  IN  EDUCATION 
IN  THE  GRADUATE  SCHOOL  OF  THE  UNIVERSITY 
OF  ILLINOIS,  1922 


URBANA,  ILLINOIS 


Digitized  by  the  Internet  Archive 
in  2015 


https://archive.org/details/anaiysisofpupilsOOmadd 


UNIVERSITY  OF  ILLINOIS 


THE  GRADUATE  SCHOOL 


June _ 2 


I HEREBY  RECOMMEND  THAT  THE  THESIS  PREPARED  UNDER  MY 

SUPERVISION  BY_^  GRACE  ERUHIIS  I-IADDEH - 

ENl'ITLED^  _ M MIALYSIS  OF  PUPILS*  lIISTAiaS  DT  GEOMSTHY 


BE  ACCEPTED  AS  FULFILLING  'PHIS  PART  OF  THE  REQUIREMENTS  FOR 
THE  DEGREE  OF  MSTER  OF  ARTS  III  3DUCATI0II 


In  Charge  of  Thesis 


Head /of  Department 


Recommendation  concurred  iiU 


Committee 


on 


Final  Examination'* 


Required  for  doctor’s  degree  but  not  for  master's 


/lopcN  n 

.J' . o’  •->'  ' / 


TABLE  OP  CQNTENIS 


Page 


Introduction 1 

Chapter  I.  Similar  Studies 2 

Chapter  II.  An  Analysis  of  the  Errors 

Tables. 

The  Nature  of  the  Errors 

Chapter  III. Causes  of  Mistalces  and  Hemedial  Suggestions 50 

Appendix,  The  Examination  Questions ....57 

Bibliography. 70 


1 


INTBODUCTION 

The  purpose  of  this  Investigation  is  to  analyze  the  difficulties  of 
high  school  pupils  in  Plane  Geometry,  by  noting  and  classifying  their  errors, 
and  to  suggest  methods  of  teaching  which  may  be  useful  in  meeting  these 
difficulties.  Examination  papers  written  January  1922  were  secured  from 
Decatur,  Clinton,  Jacksonville,  Champaign,  Urbana,  and  University  of  Illinois 
hl^  schools.  Some  semester  examination  papers  from  January  and  June  1921  were 
available  at  Champaign  High  School  and  were  included.  Quiz  papers  similar  to 
the  semester  examinations , except  that  they  were  shorter,  from  Decatur  and 
Champaign  were  used.  Four  hundred  eighty-eight  semester  papers  in  response  to 
eleven  sets  of  questions  and  117  quiz  papers  based  on  three  sets  of  questions 
were  examined. 

The  errors  were  analyzed  and  those  of  like  nature  listed  together  u.dei 
a descriptive  name.  The  papers  had  been  previously  marked  by  the  teacher  of  eac]i 
section.  His  corrections  or  suggestions  to  the  pupil,  noted  on  the  paper,  w’ere 
used  in  analyzing  the  exact  nature  of  the  error.  TJhere  no  comment  was  offered 
the  nature  of  the  error  was  determined  after  a study  of  the  question  asked  and 
of  satisfactory  responses  by  other  pupils  for  which  full  credit  was  given. 

The  errors  were  then  classified  according  to  the  mental  processes  in- 
volved and  the  per  cents  determined.  Since  the  nature  and  comparative  number  | 
of  mistakes  varied  with  the  type  of  question  asked,  a large  number  of  papers 
in  response  to  a variety  of  quest lonswere  used  and  the  per  cent  of  error  of 
each  type  compared  with  the  per  cent  of  questions  calling  for  responses  of  that 
type.  The  semester  grades  of  the  pupils  were  secured  also.  These  data  and  the 
method  of  selection,  all  papers  available  in  each  school  were  used.  Indicate  a 
normal  selection  of  errors. 


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2 


Chapter  I 


SIMILAR  STUDIES 


1.  C.W. Odell,  of  the  University  of  Illinois,^  (in  ’’School  and  Society” 


December  31,1921)  reports  ”A  Study  of  One  Thousand  Errors  in  Latin  Prose 
Composition.”  His  data  were  obtained  from  quizzes,  written,  class  and  board 
work,  of  pupils  in  a small  middle  western  town,  Pour  teachers  cooperated.  Two 
different  beginning  text  books  and  one  prose  composition  book  plus  50  per  cent 
supplementary  material,  were  used.  Errors  for  each  of  the  four  years  are 
classified  under  the  following  general  headings:  Declension,  Conjugation,  Order, 
Comparison,  Analysis,  Vocabulary,  Omissions,  and  Insertions.  The  first  two  were 
subdivided  into  Parts  of  Speech  and  under  each  the  factors  with  which  form 
varies,  for  exanople.  Case,  Declension,  Number,  Stem,  Tense,  Voice,  Person, etc. 

A second  table  gives  the  same  data  in  per  cents,  A third  table  classifies  all 
errors  into  three  groups:  (I)  errors  made  through  wrong  analysis  and  imperfect 
knowledge  of  syntax,  (II)  errors  caused  by  ignorance  of  forms,  words,  and  rules, 
which  were  chiefly  matters  of  mechanical  memory,  (HI)  errors  of  pure  careless- 
ness, A consistent  one-third  of  all  errors  was  caused  by  lack  of  mechanical 
memory.  Less  than  one-fourth  were  due  to  so-called  "lack  of  reasoning”  and  about 
one-half  to  carelessness.  No  teaching  suggestions  are  offered  although  the 
author  states  that  certain  changes  were  made  in  his  own  teaching  as  a result 
of  study  of  these  data. 


2.  Thorndike  gives  two  reports  of  studies  in  reading.  "The  Understand- 
2 

ing  of  Eentences”  is  based  on  the  answers  of  500  pupils  of  elementary  school 


1.  P.643-6. 

2.  Elementary  Educ.J. , Vol.  18, p. 98-114 


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first  year  hlgii  school  to  questions  on  material  presented  in  paragraphs,  whose 
elements  were  familiar,  but  whose  sentence  structure  was  more  elaborate  than 
pupils  of  that  grade  can  manage.  ’’The  Investigation  was  primarily  concerned  with 
ability  of  the  pupil  to  understand  totals,  few  of  whose  elements  are  untaiown,  | 
but  whose  internal  relations  are  somewhat  intricate  and  subtle."  The  pupil  is  i 
aslced  to  study  the  paragraph  and  answer  the  questions,  reading  as  often  as  he 
wishes*  Answers  typical  of  the  mistakes  in  the  order  of  their  frequency  are  | 
given.  A table  of  sixteen  overpotent  elements  and  the  responses  which  they  call 
forth  is  included  in  the  data. 

As  a result  of  these  investigations,  eight  causes  of  error  in  reading 
are  suggested. 

a.  Failure  to  set  the  mind  toward  reading  and  writing  the  answers  - 
pupil  more  interested  in  something  else  and  fails  to  try.  Such  a situation 
arises  only  rarely, 

b.  Some  other  set  of  association,  productive  of  writing,  prepotent 
over  the  "read  and  ansv«er"  set.  For  example,  a sixth  grade  pupil  copies  all  the 
questions  v^lle  another  copies  the  last  two  words  of  each  question. 

c.  Answering  the  question  may  be  prepotent,  little  or  no  reference 
being  made  to  the  paragraph.  Such  prepotency  may  effect  the  total  response  or 
it  may,  and  more  frequently  is,  only  occasional. 

d.  The  fourth  is  overpotency  of  some  element.  The  wrong  paragraph,  | 
or  a wrong  element  of  a paragraph  may  be  used,  or  a wrong  element  may  be  added  | 
to  the  correct  one.  When  a pupil  does  not  understand,  the  known  element  becomes 
overpotent, 

e.  Underpotency,  the  obvious  complement  of  overpotency,  occurs. 
Frequently  the  underpotent  element  is  a modifying  phrase  or  clause,  omission  of 
vdiich  completely  changes  the  meaning  of  the  sentence. 


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4 

f.  It  Is  also  the  case  that  elements  in  the  question  tend  to  become 
overpotent  and  to  determine  response  more  or  less  irrespective  of  the  rest  of 
the  question. 

g.  Since  three  to  four  times  as  many  gave  vrcng  answers  as  failed  to 
answer,  Thorndike  concludes  that  the  pupil  has  no  criterion  for  understanding  or 
accepting  the  correctness  of  an  answer.  He  accepts  a series  of  words  as  an 
answer  because  of  a superficial  appropriateness.  No  examination  is  felt  neces- 
sary. The  penalty  for  omission  should  be  lighter  than  for  incorrect  answers. 
Healthy  criticism,  but  not  such  as  to  frighten  the  child,  is  needed.  He  should 
be  encouraged  to  add  to  his  method  a careful  examination  to  see  if  the  question 
fits. 

i.  Correct  elements  may  be  transposed  into  wrong  relations.  Under- 
standing may  be  described  as  "thinking  things  together".  "The  contributory 
tendencies  of  each  word  and  word  group  have  to  be  right  not  only  in  nature,  but 
also  in  amount  of  potency  or  influence  of  force,  each  in  comparison  with  othera,” 

The  commonest  cause  of  error  is  underpotency  or  overpotency  of  elements 
of  the  question  or  paragraph. 

3.  "Reading  as  Reasoning;  A Study  of  Mistakes  in  Paragraph  Reading," 
similar  in  method  and  conclusions  by  the  same  author,  is  based  on  all  responses 
of  200  pupils  of  grade  six  to  four  questions  about  one  paragraph.  The  data  show 
frequency  and  per  cent  of  frequency  for  each  response  and  a table  of  ten  over- 
potent  words  with  the  answers  they  influenced. 

He  shows  that  reading  may  be  wrong  or  Inadequate  from  three  different 

causes: 

a.  Wrong  connection  with  words  singly. 

b.  Overpotency  or  underpotency  of  elements. 

3.  J.  of  Educ.  Psych. ,Vol.8,p.323-332,  June,  1917. 


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5 


c.  Pallure  to  treat  Ideas  produced  by  reading  as  provisional, 
and  80  to  inspect  and  welcome  or  reject  them  as  they  appear. 

The  relation  of  these  three  elements  of  reading  to  the  following  three 
elements  of  solving  mathematical  problems  is  pointed  out: 

a.  Selecting  right  elements. 

b.  Putting  them  together  in  right  relations. 

c.  hight  amount  of  weight  of  influence  or  force  of  each. 

Thorndike  concludes  that  reading  involves  the  same  sort  of  organiza- 
tion and  analytic  action  of  ideas  as  occurs  in  thinking  of  supposedly  higher 
sorts.  "It  appears  likely,  therefore,  that  many  children  fall  in  features 

of  other  subjects  not  because  they  have  understood  and  remembered  the  facts  and 
principles  but  have  been  unable  to  organize  and  use  them;  or  because  they  have 
understood  them  but  have  been  unable  to  remember  themj  but  because  they  never 

4 

understood  them," 

4.  In  connection  with  an  analysis  of  mental  processes  Huger’ s 
"Psychology  of  Efficiency,  k Study  of  Mechanical  Puzzles"  is  suggestive.  His 
problem  was  to  analyze  human  methods  of  meeting  relatively  novel  situations 
and  of  reducing  their  control  to  acta  of  skill.  It  deals  with  the  part  dif- 
ferent sorts  of  thought  processes  actually  play  in  meeting  novel  situations, 
and,  so  far  as  possible,  with  conditions  favoring  the  development  of  variation. 
Mechanical  puzzles  which  Involve  actual  manipulation  of  materials  were  used. 

All  were  possible  of  solution.  For  the  most  part  they  were  made  of  wire  and 
Involved  removing  of  some  part  of  the  apparatus,  such  as  a ring,  star,  or  | 

heart,  from  the  remaining  portion.  Most  of  them  were  analytical  and  tridimen- 
sional, The  movements  required  were  rather  complex. 

The  subject  examined  each  puzzle  separately  for  an  average  of  fifty 
trials  of  one  to  one  and  a half  hours  in  the  presence  of  the  examiner,  A 
4.  J.  of  Educ.Psych, ,Yol,8,p. 331. 


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■ ‘r-  . ’ i 'C-  '^i^ '•:  : iiloi 
f:'T  c.-.  ;■  ,i;  .'JcC'iftl 

* ‘i  "0‘'.  , ■ . • -tsl^  . ..r 

' '.■■  ;vr‘V;|’  njf-J'- 

• i . ' • f . ,i.  j ■.  :oq  - J A 

4 

■^'  ■'•  . . ■-!;i 'itc'-. 

t 

' I. . ji  i 'la:??-;  v:  . -i’' 

- V .'••  . ;v>  , ■ -firi* 


n • 


triple  account  was  taken  of  each  trial.  One  account  was  written  or  dictated 
by  the  subject  at  the  close  of  each  trial.  Running  accounts  of  several  subjects 
were  record®d  by  the  examiner  In  the  first  two  or  three  trials  with  a new  puzzle. 
In  each  case  the  examiner  wrote  a description  of  all  the  movements  as  they  were 
made  by  the  subject. 

The  function  tested,  analysis  In  tridimensional  space,  was  relatively 
poorly  developed.  The  subject  could  not  mentally  construct  in  any  completeness 
the  spatial  transformations  required.  Discriminations  were  difficult  since  no 
complete  system  of  terms  was  available  to  stand  for  the  discriminations  when 
once  made.  The  ideas  with  which  a subject  sought  to  reason  out  the  problem  were 
not  closely  enough  related  to  the  case  In  hand  to  be  of  much  value.  Some  were 
of  negative  value,  eliminating  one  method  as  Impossible,  others  a positive  hin- 
drance starting  a subject  on  a futile  method.  The  lack:  of  Ideas  closely  related 
to  puzzles  and  capacity  of  constructing  transformations  in  three  dimensions  was 
reflected  In  the  attitude  of  the  pupils.  Round  about  methods,  no  decrease  In 
time  for  repeated  successes,  som=tluies  an  increase,  success  the  result  of  random 
movements,  all  showing  lack  of  analysis  were  characteristic  of  the  behavior  of 
the  subjects. 

Types  of  analysis  of  the  puzzles  varied  in  explicitness  from  a vague 
feeling  of  familiarity  at  chance  recurrence  to  ability  to  use  analysis  In  a 
novel  situation  and  give  a general  formula  for  Its  use  under  varying  conditions. 
Analysis  varied  In  extent  from  a partial,  such  as  a locus  analysis  - simply  re- 
cognition of  the  part  of  the  puzzle  involved  in  an  accidental  solution  - to  a 
total  unified  analysis,  either  factual  Image  type  or  general  formula,  recognizing 
the  process  as  a single  structure.  The  relation  of  time  of  analysis  to  motor 

I 

variation  varied.  Motor  variation  may  be  successful  but  unnoticed,  or  analysis 


j 


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7 

may  come  first  and  motor  response  follow  much  later.  Analysis  and  motor  varia- 
tion may  be  simultaneous  yet  distinguishable,  or  analysis  may  occur  at  some 
point  of  the  motor  variation  and  the  course  of  the  movement  be  continued  purpose- 
ly. Both  sensory  or  perceptual  and  image  or  ideational  analysis  occurred.  The 
former  often  came  with  a rush  or  flash.  The  latter  was  advantageous  in  lopping  | 
off  irrelevant  data.  Verbal  image  has  advantage  over  factual  in  still  further  j 
foreshortening  the  process  and  in  greater  control  in  recall.  | 

Consciously  adopted  variation,  with  explicit  analysis  of  variation  was 
the  most  efficient  raethod.  Anticipatory  analysis  was  difficult  due  to  the  in- 
ability of  picturing  transformations  and  the  strong  impulse  to  manipulate. 
Greometric  knowledge  was  of  little  value.  Ideas  of  general  scientific  method 
of  procedure  seemed  to  be  of  more  importance  in  attacking  new  problems  than  ideas 
of  geometry.  The  factors  entering  into  the  problem  attitude,  confidence  and 
high  intellectual  activity  and  attention,  freedom  toward  variation,  and  critical 
evaluation  of  suggestions,  were  noted  to  be  connected  with  efficient  forms  of 
response. 

5.  Belated  to  Huger' s "Study  of  Mechanical  Puzzles"  is  earth's 
"Psychology  of  Biddle  Solution;  An  Experiment  in  Purposive  Thinking."  The  rid- 
dles required  a mental  solution  rather  than  mechanical  manipulation.  The  pur- 
pose of  the  study  was  to  compare  the  methods  of  purposive  thinking  used  in  solv- 
ing riddles  with  the  methods  used  in  learning  by  trial  and  error.  The  subjects 
of  this  experiment  were  three  hundred  and  thirty-one  college  and  normal  school 
students.  The  records  of  other  individuals  who  knew  one  or  more  of  the  riddles 
were  discarded.  Ten  riddles  were  selected  because  of  their  seeming  fairness  as 
mental  problems,  Biddles  capable  of  only  one  answer  were  used  as  far  as  possible. 
The  subject  was  allowed  three  minutes  for  each  riddle.  He  recorded  all  guesses 
both  right  and  wrong.  A sample  record  sheet  is  included  in  the  data.  One  table 
5,  J.B, Garth,  J,  of  Educ.  Psych.,  Jan.  1920,  Vol.II,  p, 16-33. 


-I 


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..■'fyf-.i.'-  v'niii':58u  'lo  D‘-.:>ijri6  t»’x.«r»?  volftail  ,t>Sl>^p'&B JJb' eatw. 

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i.i  ,7J.'o  ,'^U  .coi  .oc6»  Id  . - .?lr 


I — 


arranges  the  riddles  in  order  of  difficulty  as  based  on  frequency  of  solution. 

The  following  Is  a typical  easy  riddle.  '*If  your  aunt’s  sister  Is  not  your  aunt  , 
then  ^at  kin  Is  she  to  you?"  "What  Is  It  that  asks  no  questions  yet  requires 
to  be  answered  many  times?"  Illustrates  a hard  riddle.  The  Introspections  of 
five  subjects  on  riddle  one  and  four  subjects  on  riddle  two  show  the  associa- 
tions which  suggest  each  guess.  The  guesses  are  tabulated  to  show  frequency  forj 

I 

each  riddle.  Additional  tables  show  the  relation  of  difficulty,  frequency,  and  i 
rare  guesses. 

Garth  sets  forth  some  rather  definite  conclusions  from  the  experiment. 

(l.)  One  must  believe  In  the  trial  and  error  character  of  the  method  em- 
ployed In  riddle  solution. 

(2.)  Speedy  guessing  tends,  as  this  objectively  determined,  to  militate 
against  successful  guessing. 

(3.^  Rare  associations  characterize  speedy  guessing. 

(4.^  Usualness  or  homogeneity  of  guessing  marks  the  average  guesser’s 
association  and  is  not  characteristic  of  the  speedy  guessers. 

(5.'i  Slow  guessing  Is  not  marked  by  successful  solution  or  homogeneity  of 
associations.  However,  the  unusualness  as  determined  in  this  experiment  Is  not 
so  great  as  In  the  case  of  speedy  guessers. 

This  experiment  with  riddles  gives  evidence  for  Thorndike’s  statement,^ 
"This  process  of  attentive  consideration  and  selection  or  rejection  is  clearly 
shown  In  the  search  of  a proper  word  to  express  a meaning.  In  attempts  to  solve  | 
problems  of  all  sorts  from  the  simplest  riddle  or  puzzle  to  the  most  abstruse 
question  In  mathematics  or  science  and  in  summoning  evidence  to  support  argument!' 

n 

The  results  also  bear  out  Dewey  In  his  conclusion  that  "Too  few  sug- 
gestions, Indicate  a dry  and  meager  mental  habit On  the  other  hand,  sug- 

gestions may  be  numerous  and  too  varied  for  the  beat  interests  of  the  mental 
E.  B.T  .Thorndike,  "Elements  of  Psychology,"  p.265. 

7»  John  Dewey,  "How  We  Think."  P«36. 


. j.r‘5 ''^ar ''  ' 

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6'  ,:i^'  te:^~  <»■ 


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9 


habit,  so  many  suggestions  may  rise  that  the  person  Is  at  a loss  to  select  a- 
mong  them The  best  mental  habit  involves  a balance  between  paucity  and  re- 

dundancy of  suggestions.” 

6.  Thorndike's  ” Psychology  of  Arithmetic”  contains  much  suggestive 
material.  The  chapters  on  "The  Constitution  of  Arithmetical  Abilities”® 
analyzes  certain  important  or  neglected  abilities  of  arithmetic  as  samples. 

As  a first  example  the  meaning  of  a fraction  is  considered.  Seventeen  minor 
abilities  are  listed.  Scientific  teaching  builds  up  the  total  ability  to  manip- 
ulate with  fractions  as  a fusion  or  organization  of  these  lesser  abilities. 

At  the  same  time  that  these  minor  abilities  are  being  developed  the  pupil  gets 
his  knowledge  of  the  meaning  of  a fraction  at  zero  cost.  In  the  case  of  the 
meaning  of  a fraction,  the  ability  and  so  the  learning,  is  much  more  elaborate 
than  common  practice  has  assumed;  ”ln  the  case  of  subtraction  and  division 
tables  the  learning  is  ouch  less  so.”  Except  perhaps  in  the  case  of  the  dullest 
twentieth  of  pupils,  the  bonds  formed  in  the  subtraction  and  division  tables  j 

I 

are  somewhat  facilitated  by  the  already  learned  addition  and  multiplication. 
Instead  of  memorizing  these  tables  independently  the  pupil  may  by  a properly 
arranged  set  of  exercises  derive  them  fronl^the  corresponding  addition  and  multi- 
plication. I 

A thorough  analysis  of  the  mental  function  involved  in  arithmetical  \ 

t 

t 

learning  turns  into  the  question,  ”7/hat  are  the  elementary  bonds  that  constitute' 

ij 

these  functions?”  The  problem  of  teaching  arithmetic  is  psychologically  a j| 

! ! 

problem  in  the  development  of  a hierarchy  of  intellectual  habits,  which  becomes 

ii 

in  large  measure  a problem  of  the  choice  of  bonds  to  be  formed  and  of  the  dis- 

t 

covery  of  the  best  order  into  which  to  form  them  and  the  best  means  of  forming 
each  in  that  order. 

M 

Pupils  learn  the  method  of  manipulation  of  numbers  not  by  deduction 
8.  Psychology  of  Arithmetic,  p.59. 


‘ ^ * ly*n 

? cf  •i'E^u'C.’  li'U  s J's-  Cl  Goaietj.  ^Pil#  o«J*i 


^ - - 


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it'  « 


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; r-~  - -5,  .*A  , 

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ni 


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iq  ,p-1 


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JiU*.,<.  tr»i  nisi'i  03  rtoirfii  o4ni  ‘lOl-iO  0ri4  ’io:^\;;5i0YOo  \ jf 


o; 


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,'5iy»«Wi#ilA  tP  tljOi  0410141^  .5 


viji 


r,  ,4 


10 

from  their  taaowledge  of  decimal  notation  but  rather  by  seeing  the  method  employ- 
ed and  acquiring  them  as  associative  habits.  They  add  the  3 of  23  to  the  3 of 
53  and  the  2 of  23  to  the  5 of  53  at  the  start,  in  nine  cases  out  of  ten  be- 
cause they  see  the  teacher  do  so.  All  but  the  very  dullest  tv^entieth  or  so  of 
children  come  In  the  end  to  something  more  than  rote  knowledge,-  to  understand, 
to  know  that  the  procedure  is  right.  V/e  who  have  already  formed  and  long  used 
the  right  habits  can  hardly  realize  the  force  of  mere  association.  The  pupil 
writes  61  for  16  for  the  same  reason  that  he  writes,  63,  64,  65,  26,  36,  56,  and | 
so  on  correctly.  He  has  learned  to  write  the  6 in  the  same  order  in  which  he 
speaks  It.  If  the  pupil  has  been  drilled  In  writing  28  after  the  sum  of  8,6, 

9,  and  5,  he  will  be  sure  to  do  so  in  two  column  addition,  even  though  a second 
column  la  added  also,  unless  some  counter  force  influences  him.  Such  cases 
illustrate  the  fact  that  the  learner  rarely  can,  and  almost  never  does,  survey 
and  analyze  an  arithmetical  situation  and  justify  what  he  is  going  to  do  by  ar- 
ticulate deductions  from  principles.  He  usually  feels  the  situation  more  or  lessi 
vaguely  and  responds  to  It  as  he  has  responded  to  it  or  some  situation  like  it 
in  the  last. 

Psychologists  wish  the  pupil  to  reason  not  less  than  he  has  in  the 
9 

first  but  more.  ’’They  find,  however,  that  you  do  not  secure  reasoning  in  a 
pupil  by  demanding  it,  and  that  his  learning  of  a general  truth  without  the  pro- 
per development  of  organized  habits  back  of  it  is  likely  to  be,  not  a rational 
learning  of  that  general  truth,  but  only  a mechanical  memorizing  of  a verbal 
statement  cfjit.  The  newer  pedagogy  is  careful  to  help  him  build  up  these  con- 

I 

nections  or  bonds  ahead  of  and  along  with  the  general  truth  or  principle,  so  thali 
he  can  better  understand  it.  It  secures  more  reasoning  in  reality  by  not  pre- 
tending to  secure  so  much.  The  newer  pedagogy  of  arithmetic,  then,  scrutinizes 
every  element  of  knowledge,  every  connection  made  in  the  mind  of  the  learner, 

9.  Thorndike,  "Psychology  of  Arithmetic,"  p.73. 


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11 

so  as  to  choose  those  which  provide  the  most  instructive  experiences,  those 
which  will  grow  together  into  an  orderly,  rational  system  of  thinking  about 
ntirabers  and  quantitative  facta.  It  is  not  enough  for  a problem  to  be  a case 
of  some  rule;  it  must  help  review  and  consolidate  habits  already  acquired  or 
lead  up  to  and  facilitate  habits  to  be  acquired.  Every  detail  of  the  pupil’s 

( 

work  must  do  maximum  service  in  arithmetical  learning,"  | 

10 

Thorndike  next  lists  desirable  bonds  now  often  neglected. 

(1) ,  "In  the  case  of  all  save  the  very  gifted  children,  the  additions  with 
higher  decades  - that  is,  the  bonds,  16  plus  7 equals  23,  26  plus  7 eiguals  33, 

14  plus  8 equals  22,  24  plus  8 equals  32,  and  the  like  need  to  be  speci- 

fically practiced  until  the  tendency  becomes  generalized.  The  quotients  with 
remainders  for  the  divisions  of  every  number  to  19  by  2,  every  number  to  29  by 
3,  every  number  to  39  by  4,  and  so  on  should  be  taught  as  well  as  the  even 
divisions, 

(2) .  "The  equation  form  is  the  simplest  uniform  way  yet  devised  to  state 
a quantitative  issue.  Consequently  this  form  should  be  employed  widely  in  ac- 
counting and  the  treatment  of  commercial  problems,  since  it  is  a leading  con- 
tribution of  algebra  to  business  and  industrial  life.  Arithmetic,  hov/ever, 
can  present  it  nearly  as  well," 

(3) ,  In  multiplying  and  dividing  with  fractions,  special  bonds  should  be 
formed  to  counteract  the  nov/  harmful  influence  of  the  "multiply  = get  a larger 
number",  and  "divide  = get  a smaller  number"  bonds  which  all  work  with  integers 
has  been  reinforcing.  The  following  rules  for  multiplying  and  dividing  by  a 
fraction  will  counteract  the  old  habit. 

Mult iolication. 

When  you  multiply  a number  by  anything  more  than  1 the  result  is 
larger  than  the  number, 

10,  Thorndike,  "Psychology  of  Arithmetic",  p. 75. 

11.  Ibid, , p.79. 


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When  you  multiply  a number  by  1 the  result  Is  the  same  as  the  number. 

When  you  multiply  a number  by  anythini?  less  than  1 the  result  is 
smaller  than  the  number. 

Division. 

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■ 

than  the  number.  | 

When  you  divide  a number  by  1 the  result  is  the  same  as  the  number. 

When  you  divide  a number  by  anything  less  than  1 the  result  is  larger 
than  the  number, 

(4), Bonds  should  early  be  formed  between  manipulations  of  numbers  and 
means  of  checking,  A discussion  of  wasteful  and  harmful  bonds  follows.  We  in- 
clude one  example.  The  multiplications  of  2 to  12  by  11  and  12  as  single  con- 
nections should  be  left  for  the  pupil  to  acquire  by  himself  as  he  needs  them. 
These  connections  Interfere  with  the  process  of  learning  two  place  multiplica- 
tion. 

Be  concludes,  "When  we  have  cured  all  of  our  faults  in  respect  to 
bonds  now  neglected  that  should  be  formed  and  useless  or  harmful  bonds  formed 
for  no  valid  reason,  and  found  all  the  possibilities  for  wiser  selection  of 
bonds  we  shall  have  enormously  improved  the  teaching  of  arithmetic.  The  ideal 
is  such  choice  of  bonds  as  will  improve  the  function  in  question  at  the  least 
cost  of  time  and  effort.  The  guiding  principles  may  be  kept  in  mind  in  the  form 
of  seven  simple  but  golden  rules:- 

(1) ,  Consider  the  situation  the  pupil  faces, 

(2) .  Consider  the  response  you  wish  to  connect  with  it. 

(3) .  Form  the  bond:  do  not  expect  it  to  come  by  a miracle, 

(4) .  Other  things  being  equal,  form  no  bond  that  will  have  to  be  broken, 

12, Thorndike , "Psychology  of  Arithmetic",  p. 101. 


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(5) .  Other  things  being  equal,  do  not  form  tw5  or  three  bonds  when  one  will 
serve. 

(6) .  Other  things  being  equal,  form  bohds  in  a way  that  they  are  required 
to  act. 

(7) .  flavor,  therefore,  the  situation  T^ich  life  Itself  will  offer,  and 
the  response  which  life  itself  will  demand.” 

In  the  chapters  on  psychology  of  drill  in  arithmetic,  Thorndike  sug- 
gests certain  general  facts.  (1).  The  constituent  bonds  involved  in  the  funda- 
mental operation  with  numbers  need  to  be  much  stronger  than  they  are  now. 

These  bonds  should  be  strong  enough  to  abolish  errors  in  competition,  excdpt 
those  due  to  temporary  lapses.  (2).  Certain  bonds  are  of  service  for  only  a 
limited  time  and  so  need  to  be  formed  only  to  a limited  degree  of  strength.  The 
general  deductive  theory  of  arithmetic  should  not  be  learned  only  to  be  forgot- 
ten. What  is  learned  should  be  learned  much  later  than  now  and  should  be  among 
the  most  permanent  of  a pupil’s  stock  of  knowledge.  The  formation  of  bonds  to 
a limited  strength  because  they  are  to  be  lost  in  their  first  form,  in  order 
to  be  worked  over  in  different  ways  in  other  bonds  to  which  they  contribute  is 
the  most  important  case  of  low  permanence  of  bonds.  (3).  Bonds  and  abilities 
should  rarely  be  formed  each  by  itself  alone  and  never  kept  so.  Every  oond 
formed  should  be  formed  with  due  consideration  of  every  other  bond  that  has  been 
or  will  be  formed:  every  ability  should  be  practiced  in  the  most  effective 
possible  relation  with  other  abilities. 

I 

The  chapter  entitled  ’’The  Psychology  of  Thinking*'  presents  certain  | 
notions  of  abstractions,  generalization,  analysis,  and  reasoning  in  a manner 
suggestive  for  our  problem  (errors  in  geometry). 

Abstraction  and  generalization  depend  upon  analysis  and  upon  bonds 
formed  with  more  or  less  subtle  elements  rather  than  with  gross  total  concrete 
situations.  Three  means  are  employed  to  facilitate  analysis.  The  first  of 
====_____=^ 


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14 

these  Is  having  the  learner  respond  to  the  total  situation  containing  the 
element  In  question  with  the  attitude  of  piece-meal  examination,  and  with  alert 
ness  to  one  elenaent  after  another.  The  second  means  Is  having  the  learner  re^- 
spond  to  many  situations  each  containing  the  element  in  question  but  with  vary- 
ing concomitance.  The  third  is  having  the  learner  respond  to  situations  which, 
pair  by  pair,  present  the  element  in  a certain  context  and  present  that  same 
context  with  the  opposite  of  the  element  in  question,  or  something  very  unlike 
the  element,  A child  is  taught  to  respond  to  "one  fifth  of  a cake"  and  in  con- 
trast "five  cakes".  These  means  utilize  the  laws  of  use,  disuse,  satisfaction, 
and  discomfort,  to  disengage  a response  element  from  gross  total  responses  and 
to  attach  it  to  some  situation  element.  "What  happens  in  such  cases  is  that 
the  response,  by  being  connected  with  many  situations,  alike  in  the  presence 
of  the  element  in  question  and  different  in  other  respects,  is  bound  firmly 
to  that  element  and  loosely  to  each  of  its  concomitants.  Conversely,  any 
element  is  bound  firmly  to  any  one  response  that  is  made  to  all  situations  con- 
taining it  and  very,  very  loosely  to  each  of  these  responses  that  are  made  to 
only  a few  of  the  situations  containing  it,"  A situation  then  acquires  bonds 
not  only  with  some  response  to  it  as  a gross  total,  but  also  with  responses  to 
any  of  its  elements  that  have  appeared  in  any  other  gross  total. 

Learning  by  analysis  does  not  often  proceed  in  the  carefully  organ- 
ized way  representated  by  the  most  ingenious  marshalling  of  comparing  and  con- 
trasting activities.  The  associations  may  come  in  a haphazard  manner  over  a 
long  interval  of  time.  The  process  of  analysis  is  the  same  in  such  casual,  un- 
systematized formation  of  connections  with  elements  as  in  the  deliberately 
managed  piecemeal  inspection,  comparison,  and  contrast  mentioned  above.  The 
arrangement  of  a pupil's  experiences  may  be  "systematic"  by  fixed,  formal  exer- 
cises or  "opportunistic"  by  much  less  formal  exercises,  spread  over  a longer 


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15 


time  and  done  more  or  less  Incidentally  in  other  connections.  The  systerastic 
method  is  more  formal  and  artificial.  The  opportxinist ic  method  takes  advantage 
of  pupils*  interests  and  genuine  life  situations.  It  is  harder  to  manage  but 
more  desirable  i»dien  it  can  be  properly  administered  and  results  tested. 

We  may  expect  much  improvement  in  the  formation  of  abstract  and 

j 

general  ideas  in  arithmetic  from  the  application  of  three  additional  principles,! 
They  are:  (1),  provide  enough  actual  experiences  before  asking  the  pupil  to  I 

I 

understand  and  use  an  abstract  or  general  idea,  (2),  develop  such  ideas  gradu- 
ally, not  attempting  to  give  complete  and  perfect  ideas  all  at  once,  (3), 
develop  such  ideas  so  far  as  possible  from  experiences  vdiich  will  be  valuable 
to  the  pupil  in  and  of  themselves,  quite  apart  from  their  merits  as  aids  in 
developing  the  abstraction  or  general  notion. 

Working  with  qualities  and  relations  that  are  only  partly  understood 
does  under  certain  conditions  give  control  over  them.  The  general  process  of 
analytic  learning  in  life  is  to  respond  as  well  as  one  can;  to  get  a clearer 
idea  thereby;  to  respond  better  next  time;  and  so  on.  What  begins  as  a blind 
habit  of  manipulation  started  by  imitation  may  grow  into  the  power  of  correct 
response  to  the  essential  element,  A pupil  should  not  first  master  a principle 
and  then  merely  apply  it.  On  the  contrary,  the  applications  should  help  to 
establish,  extend,  and  refine  the  principle  - the  work  a pupil  does  with  number^ 
should  be  a main  means  of  increasing  his  understanding  of  the  principles  of 
arithmetic  as  a science,  | 

In  proportion  as  thinking  is  purposive,  with  selection  from  the  ideas 
that  come  up,  and  in  proportion  as  it  deals  with  novel  problems  for  which  no 
ready  made  habitual  response  is  available,  and  in  proportion  as  many  bonds  act 
together  in  an  organized  way  to  produce  response,  we  call  it  reasoning.  In 
both  inductive  and  deductive  reasoning  the  process  involves  the  analysis  of 

facts  into  their  elements,  the  selection  of  the  elements  that  are  deemed  sig- 
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16 

nlf leant  for  the  question  In  hand,  the  attachment  of  a certain  amount  of  im- 
portance or  'weight  to  each  of  them,  and  their  use  in  the  right  relations. 

Thought  may  fail  because  it  has  not  suitable  facts  or  does  not  select  from  them 
the  right  ones,  or  does  not  attach  the  right  amount  of  weight  to  each,  or  does  j 
not  put  them  together  properly.  f 

If  we  laclc  any  of  the  necessary  facts,  the  first  task  of  reasoning  is 
to  acquire  these  facts.  Other  things  being  equal,  problems  where  some  fact  abou: 
common  measures  must  be  brought  to  bear,  or  some  table  or  prices  or  discoimts 
must  be  consulted,  or  some  business  custom  must  be  remembered  or  looked  up  are 
somewhat  better  training  in  thinking  than  problems  where  all  the  data  are  given 
in  the  ‘problem  itself.  At  least  it  is  unwise  to  have  so  many  problems  of  the 
latter  sort  that  the  pupil  may  come  to  think  of  a problem  in  applied  arithmetic 
as  a problem  where  everything  is  given  and  he  has  only  to  manipulate  the  data. 
Life  does  not  present  its  problems  so. 

Arithmetical  problems  should  not  be  so  stated  as  to  rule  out  all 
quantitative  elements  except  those  vhich  should  be  considered.  If  they  are  the 
pupils  are  tempted  to  think  that  in  every  problem  they  must  use  all  the  quanti- 
ties given.  The  elements  selected  must  not  only  be  right  but  also  in  the  right 
relations  to  one  another. 

Reasoning  or  selective,  inferential  thinking,  is  not  at  all  opposed  to, 
or  independent  of,  the  laws  of  habit,  but  really  is  their  necessary  result  under 
the  conditions  imposed  by  man's  nature  and  training,  "It  is  true  that  man's 
behavior  In  meeting  novel  problems  goes  beyond,  or  even  against,  the  habits 
represented  by  bonds  leading  from  gross  total  situations  and  customarily  ab- 
stracted elements  thereof.  One  reason...  is  simply  that  the  finer,  subtle,  pre- 
ferential bonds,  with  subtler  and  less  often  abstracted  elements  go  beyond,  and 
at  times  against , the  grosser  and  more  usual  bonds..,.  The  other  reason  is  that 


T“'y\ 


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17 

in  meeting  novel  problems  the  mental  set  or  attitude  is  likely  to  be  one  ^diich 
rejects  one  after  another  response  as  its  unfitness  to  satisfy  a certain  de- 
siration  appears."^®  Successful  responses  to  novel  data,  associations  by  sim- 
ilarity and  purposive  behavior  are  in  only  apparent  opposition  to  the  fundamental 
laws  of  associative  learning.  Association  by  similarity  is  simply  the  tendency 
of  an  element  to  provoke  the  responses  which  have  been  bound  to  it.  Purposive 
behavior  is  the  most  important  case  of  the  influence  of  the  attitude  or  set  or 

I 

adjustment  of  an  organism  in  determining  (1)  which  bonds  shall  act  and  (2)  vdiichj 

results  will  satisfy,  Reasoning  is  not  a radically  different  sort  of  force  | 

operating  against  habit  but  the  organization  and  cooperation  of  many  habits  , ! 

thinking  facts  together*  The  pupil’s  own  total  repertory  of  bonds  relevant 

to  the  problem  is  what  selects  and  rejects.  Almost  everything  in  arithmetic 

should  be  taught  as  a habit  that  has  connections  wdth  habits  already  acquired 

and  will  work  in  an  organization  with  other  habits  to  come.  The  use  of  this 

organized  hierarchy  of  habits  to  solve  novel  problems  is  reasoning. 

14 

7.  The  purpose  of  Percival  M.  Symonds  in  his  ’’Psychology  of  Errors 
in  Algebra”  is  to  show  the  psychology  of  recurring  errors,  and  from  this  to 
point  out  methods  of  eradicating  them  or  better,  of  preventing  the  formation  of 
the  wrong  habits.  Seven  characteristics  of  behavior  are  enumerated.  The  first 
is  multiple  response  to  the  same  external  situation.  The  problem  of  teaching  | 
algebra  is  how  to  secure  the  desired  response.  A second  characteristic  of  be-  ! 
havior  is  that  the  pupil  has  a set  or  determination  to  get  the  answer.  There 
are  three  common  satisfactions  that  determine  a pupil’s  choice  of  his  answer. 

One  is  social  approval  evidenced  by  the  teacher  or  member  of  the  class.  F.eli- 
ance  on  approval  is  not  wholly  bad  but  for  success  we  must  have  some  other  means 
by  which  the  pupil  may  determine  the  answer  independently.  A second  satisfac- 
tion is  an  answer  with  which  he  may  compare  his  - the  answer  book  or  key.  An 

13.  Thorndike,  ”Psychology  of  Arithmetic”,  p,191. 

14.  The  Mathematics  Teacher,  Feb.  1922,  Vol,  15,  p.93ff. 


[ Hi  t7ioUtal4fita  4I. 


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assignment  such  as  '’Practice  these  until  you  can  get  12  right  in  six  minutes" 
will  remove  all  motive  for  misuse  of  the  answer  book.  A third  and  more  intrin- 

I 

sic  satisfaction  is  that  of  checking  the  answer. 

A third  characteristic  of  behavior  is  a practical  response  to  a 
situation.  a^+lOa  +24  = (a+8)(a+2)  is  a case  where  the  prepotency  of  getting 
8a+2a=10a  has  overshadowed  getting  a product  of  24.  The  law  of  partial  response 
works  especially  with  signs;  -4(3x-4)=  -12  x - 16.  The  effect  of  the  minus 
sign  of  the  first  4 or  the  minus  sign  of  the  second  4 is  ignored. 

A fourth  characteristic  of  behavior  is  what  Thorndilie  calls  "response 
by  analogy**.  In  any  new  situation  one  responds  as  he  would  respond  to  the 
situation  most  nearly  resembling  it  in  his  past  experience.  Nearly  all  cases 
of  unfinished  examples  can  be  traced  back  to  a previous  process  where  v&iat  is  nov 
a step  in  our  unfinished  example  was  then  the  final  result. 

x2  - 81  = 0 
( x+9 ) { x-9 ) = 0 

is  a case  of  response  by  analogy  - the  usual  response  to  x^-81.  Sometimes 
these  errors  show  how  well  previous  processes  have  been  learned.  They  surely 
show  how  v.«  have  failed  tn  emphasizing  those  elements  of  a problem  that  require 
a new  or  different  response. 

A fifth  characteristic  of  behavior  is  associative  shifting,  the  learn- 
ing process  itself.  Of  the  responses  suggested  that  one  is  selected  which  gives 
satisfaction.  3y  drill  and  repetition,  these  correct  responses  become  definite- j 

! 

ly  established  habits.  | 

Perseveration  - a tendency  to  repeat  an  act  time  after  time,  when  once  | 
it  has  been  aroused  by  some  appropriate  stimulus  - is  a sixth  characteristic  of  | 
behavior.  Perseveration  is  apt  to  occur  in  algebra  when  a pupil  is  not  working 
at  top  notch  enthusiasm  or  v;hen  he  is  disturbed  by  other  train  of  thoughts. 
Occasionally  it  may  result  from  a too  great  concentration  upon  a particular 


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feature  of  a problem.  Errors  in  vTriting,  such  as 
(2x-3)^*=  4y^  - 12  x+9 

may  be  a perseveration  from  some  previous  example.  When  one  does  all  the 
work  mentally  some  element  is  quite  apt  to  be  perseverated.  In 
y^+y®6 

y=  -3  or  -2  j 

] 

it  may  have  been  the  minus  sign  before  the  6 which  he  has  to  imagine  when  the 

6 is  transposed  or  it  may  be  the  minus  sign  of  the  3 -which  he  has  just  written,  j 

I 

I 

Anticipation  may  be  a seventh  characteristic  of  behavior.  By  this  we  \ 
mean  focusing  attention  on  some  element  ahead  of  the  writing  which  causes  the 
reaction  to  occur  before  it  should.  These  errors  are  always  accidental.  Anti- 
cipation is  slw’ays  a large  factor  in  causing  careless  errors  in  examinations. 

This  psychological  analysis  of  errors  enables  us  to  improve  our 
teaching.  First,  it  sho-ws  us  the  need  for  a psychological  analysis  of  algebraic 
processes  into  the  constituent  connection  or  bond  involved.  Especially  those 
bonds  that  require  a different  response  than  in  previous  processes  need  analysis 
in  order  to  receive  emphasis  in  future  teaching  and  drill. 

In  the  second  place,  our  analysis  shov;s  the  need  for  drill  or  practice 
in  various  processes.  Drill  or  practice  is  the  remedy  for  attention  to  certain 
elements  of  an  example  to  the  exclusion  of  certain  other  elements.  There  must 
be  differential  drill,  with  more  emphasis  on  those  connections  that  are  hindered 
by  past  responses,  and  relatively  less  emphasis  on  those  connections  that  are 
helped  by  past  responses.  Since  only  correct  responses  help  build  up  correct  j 

I 

I 

habits  many  easy  examples  should  be  given  rather  than  a few  difficult  ones. 

In  the  third  place,  since  accidental  errors  are  largely  due  to  mind 
wandering,  our  psychological  analysis  shows  need  for  reorganization  of  our  class 
room  procedure  by  timing  all  formal  activities  and  rapid  shifts  in  type  of  menta! 
processes  involved. 


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20 


We  see  mistalcea  to  be  Inconsistent  because  we  see  the  meanings  and  uses 
of  symbols  and  see  their  interrelations.  The  pupil  does  not  see  them  to  be  errors 
because  he  has  not  learned  the  meaning  of  the  symbols  and  has  not  comprehended 

I 

their  significance.  Psychology  teaches  us  that  pupils  first  learn  to  carry  out  | 
the  processes  and  manipulations,  and  thereby  acquire  the  meaning  and  significance! 
of  what  they  do. 

8.  Section  II  of  Chapter  Vi  of  Rugg  and  Clark’s  ”Scientific  Method  in 
the  Reconstruction  of  Ninth  Grade  Mathematics”  is  a detailed  analysis  of  pupils’ 
errors  as  revealed  by  the  test  given.  Table  III  in  the  appendix  gives  a com- 
plete list  of  "recurring  errors”  for  each  test,  together  with  frequency  and  per- 
centage of  occurrence. 

I 

Test  1:  Collecting  terms.  A common  error,  mistake  in  signs  in  combin- 
ing similar  terms  occurs  frequently  in  this  test.  The  error  is  due  to  lack 
of  elasticity  or  flexibility  of  attention  which  will  permit  the  pupil  to  hold  the 
result  of  one  operation  while  he  is  adjusting  to  a new  one. 

Test  2:Bvaluation  or  substitution.  Relatively  few  errors  are  made  in 
this  operation.  Nearly  one  third  are  due  to  squaring  the  product  of  literal 
factors  Instead  of  the  one  factor  designated  by  the  exponent. 

Test  3:  Simple  equations.  The  greatest  difficulty  in  solving  simple 
equations  is  illustrated  by  c = 6 as  a solution  of  4c=6c+12;  this  is  caused  by 
failure  to  hold  the  sign  of  the  -2c  in  mind. 

Test  5:  Parenthesis.  A rather  high  degree  of  frequency  has  been  secured  t 
in  the  teaching  of  this  operation.  About  one  half  of  the  mistakes  made  were  in 
the  use  of  signs. 

Test  6;  Special  products.  The  chief  difficulty  - one-fourth  of  all 
recurring  errors-ls  in  obtaining  the  sum  of  the  cross  products.  This  difficulty 
may  be  explained  by  Improper  emphasis  in  teaching  and  by  a temporary  lapse  of  at- 
tention between  the  first  and  last  terms  of  the  result  obtained. 


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21 

Test  7J  Exponents.  Addition  of  exponents  in  involution  involves  the 
greatest  difficulty.  Following  this  is  first,  (1)  the  error  of  squiring  the 
exponent  instead  of  multiplying  and  (2)  failure  to  raise  all  the  factors  of  the 
product  to  the  required  power.  Experimentation  will  tend  to  increase  marlcedly 
the  teacher’s  effectiveness. 

Test  8:  Factoring,  Thirty-nine  percent  of  all  the  recurring  errors 
Indicate  positive  inability  in  particular  types  of  factoring.  Failure  to  get 
the  correct  sum  of  the  cross  products  and  to  recognize  the  highest  common 
monomial  are  the  most  frequent.  Students  are  outstandingly  weak  in  continuing 
txie  process  until  tue  piime  factOxs  nave  ueen  founa.  The  reason  is  simply  they 
have  not  had  sufficient  practice  in  examples  requiring  the  successive  use  of 
several  factoring  processes. 

Test  9 and  10:  Fractions.  Errors  in  use  of  signs  predominate,  indicat- 
ing that  this  very  simple  process  of  collecting  signs  and  numbers  has  not  been 
properly  habituated  earlier  in  the  course. 

Test  111  Formulas.  The  error  most  in  evidence  in  this  operation  is 
that  involved  in  selecting  the  coefficient  of  the  unknown  in  the  solution  of 
fractional  equations.  The  recurrence  indicates  a lack  of  clearcut  practise  in 
naoituation  (»o  tnis  type  of  work. 

Teat  12T  Quadratic  equations.  The  sum  of  the  cross  products  diffi- 
culty occurs  here  again.  Failure  to  find  both  roots  gives  considerable 
difficulty  also. 

Test  13^  Simultaneous  equations.  Arithmetical  errors  and  mistakes  in 
signs  in  addition  and  subtraction  are  the  most  frequent. 

Test  14t  fiadicals.  Positive  inability  and  incomplete  extraction  of 


root  { V = b^  are  most  frequent  here.  The  evident  lack  of  mastery 

of  formal  skill  indicates  the  need  of  devices  to  supplement  the  text  book.  The 
teaching  emphasis  of  the  text  book  is  thoroughly  iincritical.  The  printed  prac- 


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tlce  exercise  is  a supplementary  device  for  conducting  drill  economically.  The 
fundamental  aim  of  instruction  is  the  solution  of  problems  primarily  of  the 
reasoning  or  interpretive  type.  We,  therefore,  insist  on  habituation  only  in  so 
far  as  it  contributes  to  efficiency  in  the  solution  of  such  problems, 

9,  The  purpose  of  the  study  of  Roger’s  "Tests  of  Mathematical  Ability 
and  their  Prognostic  Value"  is  to  make  an  analysis  of  the  abilities  Involved  in 
high  school  mathematics,  to  determine  their  efficiency  and  status,  their  inter- 
relation, and  also  their  connection  with  certain  other  forma  of  mental  capacity. 
Primarily  it  is  directed  toward  the  discovery  of  dynamic  and  quantitative  rela- 
tion between  mathematical  abilities,  rather  than  toward  showing  how  we  think  in 
mathematics  from  the  standpoint  of  analytic  or  structural  psychology. 

Thirteen  tests  were  selected  to  touch  as  many  forms  of  mathematical  a- 
chlevements  as  possible.  They  can  be  divided  into  three  classes,  six  tests  of 
algebraic  ability,  five  tests  of  geometrical  ability,  and  tests  of  language 
ability. 

The  subjects  were  fifty-three  girls  attending  Wadleigh  High  School  and 
sixty-one  pupils  in  Horace  Mann  School  for  girls.  The  geometrical  tests  were  as 
follows;-  A geometry  test  of  six  exercises  requiring  numerical  solution  or  proof, 
tests  of  superposition,  symmetry,  matching  solids  and  surfaces,  and  geometrical 
definitions. 

The  results  show  that  algebra  and  geometry  demand  activities  of  dif- 
ferent kinds,  although  algebraic  and  geometrical  abilities  are  positively  related 
as  is  usual  in  the  case  of  desirable  traits.  The  experimental  evidence  obtained 
suggests  that  a marked  degree  of  the  power  to  analyze  a complex  and  abstract 
situation  and  to  seize  upon  its  implications  is  the  most  indispensable  element 
in  mathematical  proficiency.  The  closeness  of  relationship  between  mathematical 
ability  and  ability  with  words  points  out  that  in  later  mathematics  as  in  primary 
arithmetic,  the  problem  of  teaching  children  to  reason  is  largely  a matter  of 


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23 


teaching  them  language  and  how  to  use  it.  Reasoning  in  school  problems  has  far 
more  to  do  with  language  involved  in  a problem  than  with  number  or  combination 
of  numbers, 

Miss  Rogers  concludes  that  these  tests  are  valuable  for  general  diag- 
nosis and  prognosis  of  individual  ability  in  mathematics, 

10,  Minnick’s  study,  "An  Investigation  of  Certain  Abilities  Pundamenta] 
to  the  Study  of  Geometry",  is  limited  to  an  investigation  of  certain  fundamental 
abilities  Involved  in  the  demonstration  of  theorems.  Corresponding  to  the  four 
steps  in  a demonstration  are  the  four  mental  abilities  with  which  this  study  is 
concerned;  namely, 

fl.)  The  ability  to  draw  a figure  for  a theorem, 

f2.)  The  ability  to  state  the  hypothesis  and  conclusion  accurately  in 
terms  of  the  figure. 

The  ability  to  recall  additional  Imown  facts  concerning  the  figure, 


and 

('4.^  The  ability  to  select  the  necessary  facts  and  to  arrange  them  so  as 
to  produce  a proof. 

The  purpose  of  this  investigation  is  threefold: 

(l,^  To  determine  the  relation  of  each  of  these  four  abilities  to 
teachers*  marks, 

(2,)  To  determine  the  extent  to  vdilch  these  abilities  are  developed  in 
our  high  schools. 

(3.)  To  develop  tests  which  may  be  used  for  the  purpose  of  diagnosis,  of 
determining  whether  or  not  the  weaknesses  of  a class  are  due  to  lack  of  develop- 
ment of  one  or  more  of  these  abilities, 

Some  of  her  conclusions  are  of  Interest.  Since  we  do  not  know  that 
the  test  measures  the  respective  abilities  in  the  same  way,  we  can  not  compare 
the  results  of  the  different  tests.  We  can  compare  the  results  of  giving  the 


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24 

same  test  in  different  schools.  In  the  case  of  each  test,  the  marks  of  some 
schools  are  quite  satisfactory  while  those  of  others  are  extremely  low.  Thus, 
although  it  is  possible  to  develop  the  abilities  v;ith  which  their  study  is  con- 
cerned, some  schools  fail  to  do  so.  k diagnosis  such  as  these  tests  provides 
will  enable  the  teachers  to  check  the  development  of  each  ability  and  give  atten- 
tion to  particular  phases  of  the  subject  as  needed. 

11.  Judd  devotes  one  chapter  of  ’’The  Psychology  of  High  School  Subjects 
to  "The  Psychological  Analysis  of  Geometry."  The  material  for  this  analysis  is 
procured  from  three  sources:  first,  a text  bock;  second,  class  recitation  by 
the  pupils;  and  third,  books  on  the  teaching  of  mathematics.  The  text  book  is 
considered  first.  In  the  presentation  of  definitions  three  processes  are  noted. 
They  are  analysis,  or  distinguishihg  the  parts,  abstraction  or  ability  to  go  be- 
yond the  real  experience,  and  use  of  symbols.  One  tries  by  the  use  of  an  ab-  I 
street  idea  to  get  rid  of  as  much  sensory  content  as  he  can  in  order  to  leave  be- 
hind the  pure  special  elements  of  the  experience. 

Jbllowing  the  definitions  in  the  text  comes  single  experimentation  and 
experimentation  through  which  the  properties  of  figures  and  geometrical  elements 
are  discovered.  The  pupil  gains  very  little  from  these  definitions  and  simple- 
analyses  until  he  has  used  them  repeatedly  in  later  proof. 

An  angle  is  used  to  illustrate  the  development  of  ideas.  The  angle  is 
first  explained  by  considering  one  single  definite  specimen  as  "the  opening  be- 
tween two  straight  lines."  Later  he  presents  the  angle  in  its  general  special 
relations  as  formed  by  rotating  a line  about  a point. 

Superposition  and  symmetry  are  introduced  as  methods  of  comparison  in- 
dispensable for  further  mental  development  and  purposes  of  application.  Symmetry 
is  not  so  direct  as  superposition.  The  former  offers  more  possibility  for 
error.  Geometry  does  not  consider  it  as  reliable  as  superposition. 


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25 


Geometry  Is  more  than  the  art  of  seeing  space.  The  student  coaipares  a 
few  simple  figures  and  then  generalizes  his  findings  in  the  form  of  a principle. 

He  must  have  enough  space  perception  to  furnish  the  basis  for  his  geometrical 
analyses  and  generalizations.  The  emphasis  on  axioms  as  opposed  to  postulates 
illustrates  the  frequent  neglect  of  space  and  the  emphasis  of  logic  by  geometri- 
cians. 

A demonstration  differs  psychologically  and  pedagogically  from  a defi- 
nition. The  latter  is  a close  compact  statement  while  a demonstration  is  a de- 
tailed and  explicit  unravelling  of  the  situation,  Harly  theorems  train  the 
student  m the  use  of  a form  of  analysis  not  demanded  in  definitions.  Combining 
the  results  of  successive  steps  of  analysts  in  the  later  theorems  calls  for  a 
"type  and  reach  of  attention  which  is  higher  than  that  required  in  learning  a 
definitioru'*^® 

In  an  original  exercise  a student  must  first  analyze  the  situation 

1 

which  is  propounded,  and  then  he  must  be  able  to  draw  out  of  his  fund  of  accumulaj 
ted  experiences  the  principles  viiich  will  help  him  in  solving  his  complex  present 
experience.  The  only  way  to  teach  a student  to  solve  originals  is  to  teach  him 
how  to  analyze  a new  problem  and  how  to  seek  among  his  store  of  experiences. 

Prom  a presentation  of  more  than  one  solution  of  a problem  and  a discussion  of 
the  merits  and  demerits  of  each,  the  pupil  will  gain  an  insight  into  the  methods 
employed  by  the  author  in  solving  the  problem. 

In  a class  recitation  the  reactions  noted  were  infinite  in  their 
variety.  Several  types  of  difficulty  arose.  A lack  of  appreciation  of  the  de- 
mands of  logic  prevented  the  class  from  arranging  a demonstration  when  all  the 
facts  were  known.  In  another  case  a student  was  not  able  to  pass  from  a logical 
demonstration  to  the  spacial  fact;  still  another  had  his  mind  fixed  on  a rule 
of  procedure  somewhat  removed  from  the  fundamental  fact.  He  was  committed  to 
15.  Judd,  "Psychology  of  High  School  Subjects",  p. 61. 


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26 


the  practical  rule  and  could  not  get  a point  of  departure  suitable  to  the  proof. 

We  have  noted  three  spheres  of  experience,  each  of  -rhich  is  so  different 
from  the  other  that  transition  out  of  one  into  another  is  difficult.  They  are, 
first,  space  perception;  second,  an  abstract  system  of  logical  steps  constituting 
proof;  and  third,  rule  of  practical  procedure.  The  student  aiust  be  trained  to 
pass  from  one  to  the  other.  There  should  be  a conscious  effort  to  view  the  matter 
from  each  of  the  possible  points  of  view  and  to  pass  from  one  to  another.  The 
student  can  see  that  the  fact  which  is  essential  for  demonstration  is  often  very 
different  from  the  point  of  highest  attention  to  the  practical  operator  or  to  the 
observer  of  space. 

Students  prefer  to  go  back  to  simple  fundamental  theorems  rather  than 
to  use  a recently  proved  theorem.  They  have  more  confidence  in  simple  theorems. 

The  part  of  the  figure  on  which  attention  was  last  concentrated  is  likely 
to  be  the  starting  point  of  the  next  step  in  observation  and  reasoning.  Therefore 
preparation  is  of  great  importance. 

All  through  the  lesson  pupils  were  called  on  to  use  their  memory  and  to 
select  from  the  propositions  and  facts  stored  up  in  memory.  The  criticisms  of 
memory  are  directed  not  against  memory  but  against  bad  forms  of  remembering.  To 
a well  ordered  body  of  knowledge  especially  if  it  is  well  ordered  in  many  directicni 
there  can  be  no  objection.  Such  a question  as  ,'*What  do  you  know  about  equilateral 
triangles?"  helps  students  to  classify  material.  Ideas  must  be  put  into  the 
mind  with  their  possible  relationships  clearly  recognized  if  these  relationships 
are,  at  some  later  day,  to  be  used  productively  in  calling  out  the  ideas. 

Reasoning  involves  memory  and  classification  of  experiences  and  the  com- 
bining of  the  experiences  which  belong  together  in  leading  to  a definite  conclusion, 
Some  general  maxims  help  the  teacher  to  induce  this  process  in  students.  "First 
try  to  foresee  in  a general  way  what  end  you  want  to  reach.  Next  marshal  all  the 


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27 


facts  you  can  find  which  are  related  to  this  end*  Then  arrange  these  facts  in  a 

1 fi 

progressive  series." 

Judd  closes  his  discussion  of  geometry  with  a comparison  of  two  hooks 
on  methods  of  teaching  geometry,  one  making  no  attempt  to  deal  with  the  psycho- 
•w  logy  of  the  situation,  the  other  colored  by  psychological  Interest  and  discusslom  , 


16.  Judd,  "Psychology  of  High  School  Subjects"  p.73 


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• ‘ vUC'-.  ^ -’i-.  -•■'  -.^v  .-rai. '• 

#*  t ■ ■ - « - ■ ’ ' ' vV-i  M jiiia#  ^ 


f '■ 


.eww«f 


28 


Chapter  II. 

AIT  AIIALY5IS  OP  PUPILS'  EHRORS. 

Table  I classifies  the  errors  noted  according  to  the  type  of  responses 
required,  giving  both  frequency  and  per  cent.  The  total  per  cent  for  each 
tjrpe  is  computed  on  the  grand  total  of  errors.  For  example,  564  or  14.1  per 
cent  of  the  total  3989  errors  occurred  in  response  to  questions  requiring  con- 
structions. The  miscellaneous  group  includes  errors  in  arithmetic,  algebra, 
grammar,  spelling,  form,  or  error  due  to  carelessness  having  little  relation 
to  the  type  of  response  required. 

Table  II  classifies  the  data  of  table  I according  to  the  mental  pro- 
cesses involved.  As  in  Table  I,  the  total  per  cent  for  each  subdivision  is 
based  on  the  grand  total  of  errors.  For  exami)le , 886  or  22,2  per  cent  of  the 
3989  errors  involved  memory,  7hen  an  analysis  of  the  error  failed  to  give  any 
cue  to  its  cause,  it  v/as  included  in  the  miscellaneous  list.  For  example,  a 
fact  wrongly  assumed  by  construction  may  have  been  the  result  of  incorrect 
analysis  of  cons tiuct ion,  of  a failure  to  reco^mize  the  logical  consequences  of 
the  construction,  or  of  oarelessness  . The  cause  could  not  be  determined  from  the 
pupil's  record  on  the  paper.  So  also,  assuming  a theorem  not  yet  proved  may 
result  fron  failure  to  recall  the  order  of  proofs  given  in  the  text  book,  or 

» 

from  a non-critical  acceptance  of  an  idea  which  is  suggested  by  the  question. 

The  pupil's  response  failed  to  indicate  v/hich. 

In  Table  III  the  questions  asted  are  classified  according  to 
the  nature  of  response  required.  Frequency  and  per  cents  are  given  for  each 
type  of  response.  V/here  a question  was  composed  of  two  or  more  parts,  each  part 
was  counted  as  a separate  question  since  it  called  for  a distinct  response.  The 


I 


f 


t 


I 


f 


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.'^  * i,' 

, ‘|.■■l:  *’  r-^vx«iv^i  68i5S 

■ vt;i.>*.  !•.*  *i.-  .tf  '■/  ,.>^:'-.0  £?'X'; 


‘I  1 1 4 - f- 


r cy  . r-ilJ- 


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*^r  v;i  ■ rto-  « * * 


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» 


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29 


total  frequency  is  therefore  not  identical  with  the  sum  of  the  questions  of  the 
examinations.  Column  3 giveg,for  comparison,  the  per  cent  of  errors  from  Table 

1 for  each  type  of  response.  The  second  entry  in  column  3 is  the  total  for 
all  proofs  both  theorems  and  exercises,  and  should  be  compared  with  the  sum  of 

2 and  3 in  column  2.  In  the  same  v/ay,  the  last  entry  in  column  3 should  be  com- 
pared with  the  sura  of  5 and  6 in  column  2. 

Table  17  gives  the  semester  grades  of  477  pupils  whose  papers  were  used 
in  this  investigation.  The  grades  of  the  pupils  of  one  section  were  not  re- 
ceived. "D”  represents  lov;  passing"  .,orh  in  most  of  the  schools.  represents 

grades  from  70  to  74  which  is  pass  ing  in  one  of  the  schools  and  conditional  in 
another.  In  the  remaining  schools,  represents  all  failures,  74  and  belov/. 
Grades  from  70  to  74  are  not  differentiated.  Several  of  the  schools  exempt  from 
final  examination  pupils  whose  average  reaches  a certain  standard.  The  best 
pupils  from  those  schools  are  excluded  from  this  study  because  of  the  source  of 
data.  Only  one  school  out  of  this  group  has  regular  raid-year  promotions,  there- 
fore, the  majority  of  the  pupils  enrolled  in  Geometry  II  during  the  first  semestea: 
from  which  the  majority  of  these  data  were  secured,  are  irregulars,  probably 
having  flunked  at  least  one  semester  of  mathematics.  This  fact  is  an  important 
factor  in  explaining  the  higher  percentage  of  low  grades  in  Geometry  II. 

Table  III  raa]©s  clear  the  fact  that  the  per  cent  of  errors  in  proofs  of 
exercises  and  theorems  is  out  of  proportion  to  the  percent  of  questions  requiring 
proof.  It  v;as  impossible  to  detemine  in  some  cases  which  statements  requiring 
proof  had  been  presented  to  the  pupil  as  theorem,  which  he  had  studied  (proofs 
of  which  he  would  be  required  to  recall  or  work  out  again),  and  7/hich  were 
exercises  that  the  pupil  may  or  may  not  hare  seen  before.  In  classifying  the 
question  any  statement  which  is  given  in  any  well  kaov/n  text  as  a theorem  or 
corollary  was  so  counted.  Others  were  termed  exercises  requiring;  proof,  "hen 


# 4 (. 


m 


.■iTv-r- 


!S  ^ 


^s. 


-,v 


■j  "n  ' 


n 'uvU  , i r ? h w.i  N.a  iR«'J:»*30rf4  ;‘ll'&i'  ,': 


. * 


1'  I 


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I’Cj  »• 


<iftu  :r  ‘iO't  C . . 

. ^ 0-  /foj^  t ?! 

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■ ■ : '^y  f*  ' ' 

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* r f 


30 

an  exercise  required  both  numerical  solution  and  proof  it  ^s  classed  with  the 
former. 

The  percents  of  error  in  construction,  numerical  exercises,  and  quo- 
tation are  somewhat  or  at  least  slightly  below  the  corresponding  percents  of 
questions.  The  comparatively  low  percents  of  errors  in  constructions  and  num- 
erical exercises  indicate  that  the  pupil  is  more  successful  in  answering  those 
types  of  questions.  Several  factors  may  tend  toward  their  success.  Construc- 
tions and  numerical  exercises  are  concrete,  and  more  closely  related  to  exper- 
ience outside  of  school,  and  therefore  of  more  Interest  to  the  average  pupil. 
Numerical  exercises  (many  of  them  very  similar  to  mensuration  problems  of  gram- 
mar school  arithmetic  or  to  those  algebraic  problems  requiring  application  of  a 
principle  or  fornaulae)  Involve  elements  in  which  the  pupil  had  received  previous 
training.  The  outstanding  difficulty  in  such  exercises  is  failure  to  analyse, 
that  is,  to  plclc  out  the  various  factors  presented,  to  reorganize  them  if  neces- 
sary so  as  to  make  clear  the  connection  between  the  required  result  and  the  I 
principle  or  formula  which  the  pupil  has  learned.  The  close  correspondence  of 
error  to  requirement  in  quotation  indicates  that  memory  for  a fact  called  for 
directly  is  a factor  but  not  a dominating  factor  in  influencing  failure. 


t) 


1.  r, 

J ' ii.  -.r."  ‘ I'.-,  •,  ■ •...  c r,-..  1 u\. uri  deici-^XG  nn 

. * : ”C-  ^ ‘ i -i'.  iiSlii: 

^ •,.(■-  •.  ■ il  , ,.;w  1 3C-:;.  c 

I " J i •"  : ' ' , . 1 j-5  - r:  J ’ i)  i - $J " ; 

'>■  . ■:  ; : I lee’tK  , -;/:.'>i;fceop  lO  r.v/;v  • 

j"  'i  . ■ ^ ...:  . :-f.  , 7:‘v:  :>c  iiOj'tertri 

-i‘^i  i,'  J ’■■■  OUC.  ' 0 «rr*]'a?uO 

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c ..  vJVlc-v/ii  la  ••iqi-onl 

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1 

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31 

Table  I 

Classification  of  Errors  According  to  Type  of  Response  Required 

in  Order  of  Frequency. 

Proofs  Frequency 

Percents 

1.  Wrong  authority 

457 

20.0 

2.  Absolute  inability 

353 

15.5 

3.  Omission  of  authority 

296 

13.0 

4.  Omission  of  step 

296 

13.0 

5.  Wrong  step 

271 

12.3 

6.  Wrong  recall  of  plan  of  proof 

108 

4.7 

7.  Wrong  conclusicn 

97 

4.2 

8.  Failure  to  recall  plan  of  proof 

82 

3.6 

9.  Omission  of  proof  of  construction 

68 

3.0 

10.  Misinterpret  hypothesis 

54 

2.4 

11.  Pact  wrongly  assumed  by  hypothesis 

47 

2.1 

12.  Omission  of  part  of  construction 

47 

2.1 

13.  Wrong  hypothesis 

45 

2.0 

14.  Use  of  theorem  to  prove  itself 

25 

1.1 

15.  Unnecessary  step 

12 

.5 

16.  Special  case 

8 

.3 

17.  Step  out  of  order 

6 

.2 

18.  Assume  theorem  not  yet  proven 

19.  Substitution  of  quantity  for  one  not 

5 

.2 

equal 

3 

.1 

20.  Example  instead  of  proof 

2 

.1 

21.  Wrong  assumption  in  indirect  proof 

1 

.0 

Totals 

2284 

57.2 

Construction 

1.  Failure  to  analyze  construction  correctly 

148 

26.2 

2.  Omission  of  explanation  of  construction  121 

21.4 

3.  Omission  of  part  of  construction 

56 

9.9 

4.  Wrong  construction  lines 

5.  Failure  to  recall  necessary  construe- 

52 

9.2 

tion  lines 

49 

8.7 

6.  Fact  wrongly  assumed  by  construction 

41 

7.2 

7.  Failure  to  recall  correct  method 

26 

4.6  1 

8.  Vague  explanation  of  construction 

20 

3.6 

9.  Inaccurate  freehand  drawing 

18 

3.3 

10.  Special  case 

15 

2.7 

11.  Complete  inability  to  construct 

11 

2.0 

12.  Omission  of  statement  of  construct  lor 

6 

1.0 

13.  Unnecessary  construction 

1 

.2 

Totals 

564 

14.1 

M .1 


32 


Exercise 8 


I. 

Flallure  to  analyze  correctly 

289 

61.4 

2. 

Wrong  interpretation  of  ’’required'* 

61 

12.7 

3. 

Failure  to  recall  formula  correctly 

38 

8.1 

4. 

Misinterpretation  of  problem 

32 

6.8 

5. 

Wrong  Interpretation  of  ’’given" 

23 

4.9 

6. 

Omission  of  proof  of  exercise 

14 

3.0 

7. 

Omission  of  part  of  exercise 

6 

1.3 

8. 

Complete  inability 

8 

1.7 

Totals 

471 

12.0 

Miscellaneous 

140 

3.5 

Grand  Total 

3989 

100.0 

iiun  mw 


r 


j 

•T 


8f*t:  ji 


Cc  A 

r :. 


p.'^ 

i:r. 

“t  C 


';i  -I  '1  or  sztitc.&  (ffi . 02ci  ijgPf. ,' . J 

*•  " -iv  , . S'OC^^  •'=. 

- ■ £ * 4 • i ^ *v  * ;j^  ' T Ii4?B6’x  0^  ••xril'M'.B 

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- ^ 


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To  Icqtsq-  to  RCl&aix^  \*  t ')/ 

- - ■■  •- 


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4 :i’’Kjil.  \rit.J  ^t>*J 
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33 


Table  II 

Classification  of  Errors  According  to  Mental  Processes 
Error 

Memory Frequency  Percent 


1.  Failure  to  quote  theorem  correctly 

325 

36.7 

2.  Failure  to  recall  theorem 

150 

16.9 

3.  Failure  to  recall  plan  of  proof  correctly  108 

12.4 

4.  Failure  to  recall  plan  of  proof 

82 

9.2 

5.  Failure  to  recall  cons’t  lines  correctly 

52 

5.9 

6.  Failure  to  recall  necessary  construction 

lines 

49 

5.' 5 

7.  Failure  to  recall  definition  correctly 

42 

4.8 

6.  Failure  to  recall  formula  correctly 

38 

4.4 

9.  Failure  to  recall  correct  method  of 

construct  ion 

26 

2.9 

10.  Failure  to  recall  definition 

13 

1.5 

11.  Unnecessary  construction 

1 

.1 

Totals 

886 

22.2 

inalysis 

1.  Failure  to  analyze  exercise  correctly 

289 

38.6 

2.  Failure  to  analyze  construction 

148 

19.8 

3.  Conclusion  wrong 

97 

13.0 

4.  Wrong  interpretation  of  "required”  in 

exercise 

61 

8.1 

5.  Misinterpret  hypothesis 

54 

7.2 

6.  Hypothesis  wrong 

45 

6.0 

7.  Misinterpret  problem 

32 

4.3 

8.  Wrong  interpretation  of  "given"  in 

exercise 

23 

3.1 

Totals 

749 

18.8 

Illogical 

1.  Use  of  theorem  to  prove  itself 

25 

58.1 

2.  Unnecessary  step 

12 

27.9 

3.  Step  out  of  order 

6 

14.0 

Totals 

43 

1.1 

•,  f 


si,.'  . I 


6 


34 

Omleelon 

1.  Omission  of  step 

296 

31.8 

2.  Omission  of  authority 

296 

31.8 

3.  Omission  of  explanation  of  construction 

121 

13.0 

4,  Omission  of  proof  of  construction 

68 

7.3 

5.  Omission  of  part  of  construction 

56 

6.0 

6.  Omission  of  part  of  proof 

47 

5.1 

7.  Vague  explanation  of  construction 

20 

2.2 

8.  Omission  of  proof  of  exercise 

14 

1.5 

9.  Omission  of  part  of  exercise 

6 

.6 

10*  Omission  of  statement  of  construction 

6 

.6 

Totals 

930 

23.3 

Ibaolute  Inability 

1,  Complete  inability  to  prove 

353 

95.0 

2.  Complete  inability  to  construct 

11 

2.9 

3.  Complete  Inability  to  work  exercise 

8 

2.1 

Totals 

372 

9.3 

Miscellaneous 

!•  Wrong  authority 

457 

45.3 

2.  Wrong  step 

271 

26.9 

3.  Carelessness 

60 

5.9 

4.  Pact  wrongly  assumed  by  hypothesis 

47 

4.6 

5.  Arithmetical  error 

44 

4.4 

6.  Pact  wrongly  asstimed  by  construction 

41 

4.1 

7.  Algebraic  error 

22 

2.2 

8.  Preehand  drawing 

18 

1.8 

9*  Special  case, construction  problem 

15 

1.5 

19*  Special  case,  theorem 

8 

.8 

11.  Spelling 

6 

.6 

12.  Vague  meaning 

6 

.6 

13*  Assumed  theorem  not  yet  proved 

5 

.5 

14.  Quantity  substituted  for  one  not  equal 

3 

.3 

15*  Exaoqple  instead  of  proof 

2 

.2 

16.  Wrong  description  of  consti*uction 

1 

.1 

17.  Wrong  assumption  in  Indirect  proof 

1 

.1 

18.  Work  in  bad  form 

1 

.1 

19.  Error  in  grammar 

1 

.1 

Totals 

1009 

25.3 

Grand  total 

3989 

100.0 

I 


35 


Table  III 


Classification  of  Questions 

Frequency 

Per  cent 

.Errors 

Construction 

33 

24.3 

14.1 

Proof  of  theorem 

32 

23.5 

57.2 

Proof  of  original  exercise 

18 

13.2 

Numerical  exercise 

35 

25.7 

12.0 

Quotation  of 

theorem 

16 

11.8 

13.2 

Definition  of 

terms 

2 

1.5 

Totals 

136 

100.0 

Table  IV 

Semester 

Grades  of  Pupils 

Orade 

Geometry  One 

Geometry  Tvo 

Frequency 

Per  cent 

Frequency 

Percent 

A 

27 

8.3 

6 

5.2 

B 

59 

18.0 

22 

14.3 

C 

72 

22.2 

23 

15.0 

D 

69 

21.3 

50 

32.5 

S 

26 

8.3 

13 

8.4 

F 

71 

22.0 

38 

24.6 

Total 

323 

100.0 

154 

100.0 

V 


..  1 


i‘S 


■■Xi 


; ■ not^efouO 


I 


o 


0 


i 


m 


36 


The  remainder  of  this  chapter  will  he  devoted  to  a consideration  of 
the  natxire  of  each  error  as  listed  In  Table  I.  Errors  in  construction  are  dis- 
cussed first,  In  the  order  of  their  frequency. 

S 

The  nature  of  the  error  in  construction. 

1.  Failure  to  analyze  construction  correctly.  Given  side  b,  an^le  B, 

and  altitude  to  side  c.  Required  to  construct  triangle  ABC.  Two  errors  In 

analysis  occur  here.  The  pupil  sometimes  does  not  know  or  falls  to  note  the 

significance  of  ’’given  altitude  to  side  c"  and  draws  the  altitude  to  side  a or 
b.  Or  he  may  sketch  the  triangle  correctly  but  fail  to  realize  that  he  has 

given  the  hypotenuse  and  leg  of  a right  triangle,  under  which  conditions  he  has 

previously  learned  how  to  construct  a right  triangle.  Continued  use  of  the  tri- 
angle notation  will  Increase  its  meaning  for  the  pupil.  Repeated  experience 
with  varied  construction  dependent  upon  "a  known  right  triangle"  will  develop  a 
habit  of  analyzing  the  situation  to  discover  "a  known  right  triangle." 

2.  Omission  of  explanation  of  construction.  Several  factors  are 
responsible  for  the  high  frequency  of  such  omissions.  The  question  did  not 
always  ask  for  an  explanation  of  a construction  when  the  teacher  felt  it  neces- 
sary. In  such  cases  the  practice  varied  widely  in  different  sections.  Some 
classes  were  accustomed  to  give  complete  explanations,  otheis  apparently  felt 
nothing  except  construction  was  necessary.  A careful  statement  of  response  re- 
quired would  probably  have  greatly  reduced  the  frequency  of  these  omissions. 

For  such  errors  the  teacher  Is  more  to  blame  than  the  pupil. 

3.  Omission  of  part  of  construction.  This  is  typically  a case  where 
the  pupil’s  attention  Is  concentrated  on  one  factor  of  the  total  response. 

That  factor  may  be  prepotent  because  of  Its  position  in  the  solution  - first 
step  - or  its  familiarity  to  the  pupil.  His  attention  does  not  include  the 


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37 


other  requirements  of  the  problem  probably  because  he  has  not  had  practice  In 
using  these  elements  together.  The  pupil  who  is  given  the  perimeter  and  asked 
to  construct  an  equilateral  triangle  divides  the  perimeter  into  three  equal 
parts  and  steps  as  he  did  when  learning  to  divide  a segment  into  a given  number 
of  equal  parts.  For  the  pupil  who  has  had  experience  in  constructing  a rectanglt 
equal  to  two-thirds  or  three-fourths  of  a given  square,  the  habitual  response 
may  be  prepotent.  He  falls  to  attend  to  the  two-thirds.  Developing  abilities 
in  the  forms  and  relations  in  which  we  wish  to  use  them  will  avoid  this  error. 

Note.  We  will  discuss  number  5 next,  and  then  4. 

5.  Failure  to  recall  necessary  construction  line.  This  error  refers 
to  the  construction  of  the  figure  required  for  the  proof  of  a theorem.  A 
typical  case  la  in  the  proof  of  ”The  sum  of  the  angles  of  a triangle  is  equal 
to  two  right  angles.**  Two  constructions  are  possible,  (1)  extend  one  side  to 
form  an  exterior  angle,  or,  (2)  construct  through  one  vertex  a parallel  to 
the  opposite  side.  Another  frequent  failure  occurs  in  the  proof  of  **If  two 
straight  lines  are  parallel  the  alternate  interior  angles  are  equal.”  The  line 
forming  alternate  Interior  angles  that  are  equal,  if  the  given  angles  are  as- 
sumed unequal  is  omitted.  Such  lack  of  recall  is  accompanied  by  failure  to 
recall  the  plan  of  proof  and  results  in  either  complete  inability  to  prove  or 
a wrong  plan.  Special  attention  is  necewsary  to  strengthen  the  bonds  between 
theorem,  figure,  and  method.  Rapid  fire  review  at  the  blackboard  may  be  used 
to  exercise  the  desired  bonds.  The  pupil  may  sketch  the  figure  with  all 
aul illary  lines  for  each  theorem  stated.  Such  exercise  will  probably  show  a need 
for  a detailed  analysis  of  the  construction  necessary  for  the  proof  of  certain 
theorems  (especially  for  contrast  and  comparison  of  similar  figures),  followed 
by  additional  drill. 

4.  Wrong  construction  lines.  The  figure  for  a theorem  and  the  one  for 


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38 


Its  converse  are  sometimes  confused.  The  construction  required  for  the  indirect 
proof  that  "Two  lines  are  parallel  if  the  alternate  interior  angles  are  equal," 
is  sometimes  given  for  its  converse  and  vice  versa.  This  is  another  case  where 
care  is  necessary  to  connect  the  correct  response  to  the  situation  and  keep 
the  bonds  strong  by  use. 

6.  Pact  wrongly  assumed  by  construction.  Sometimes  a line  is  con- 
structed to  satisfy  several  different  conditions,  one  or  two  of  which  it  may 
satisfy,  but  all  of  which  we  are  not  justified  in  assuming  that  it  can  satisfy 
at  one  time.  A common  example  is  "To  construct  a line  through  a given  point 
perpendicular  to  and  bisecting  a given  line."  Two  of  these  conditions  a line 
may  satisfy  but  all  three  can  be  true  of  one  line  only  in  very  special  cases. 

One  such  error  of  high  frequency  occurred  in  proving  that  a circle  may  be  con- 
structed through  three  points  which  are  not  in  a straight  line.  The  text  used 
by  several  of  these  classes  suggests  that  the  point  used  as  the  center  of  the 
required  circle  must  be  proved  equally  distant  from  the  three  given  points,  but 
leaves  the  reason  to  the  pupil.  The  circle  at  once  suggests  that  all  radii  are 
equal.  If  the  book  suggested  or  the  pupil  considered  that  he  is  to  prove  that 
the  circle  passes  through  all  three  points,  he  would  realize  he  had  used  as 
given  the  very  construction  he  is  to  prove  true.  Special  preparation  for  prov- 
ing construction  is  necessary  in  order  to  overcome  the  pupil's  previous  set,- 
assumlng  all  constructions  true  by  construction. 

7.  Failure  to  recall  correct  method.  "Required  to  construct  a mean 
proportional  to  m and  n."  The  pupil  remembers  the  first  step  -to  construct  a 
semi-circle  with  m plus  n as  diameter.  He  falls  to  recall  a second  step  - 
construct  a perpendicular  to  the  diameter  at  intersection  of  m and  n.  The 
attempted  proof  of  the  construction  shows  that  the  pupil  has  an  inadequate 
notion  of  proportion  and  an  Incorrect  conception  of  the  meaning  of  mean 


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39 

proportional.  A careful  explanation  of  the  nature  of  the  mean  proportional  and 
additional  practice  in  its  construction  will  avoid  such  error,  '’Inscribe  a 
circle  in  a triangle,’*  The  perpendicular  bisectors  of  the  sides  of  a triangle 
are  constructed,  to  locate  the  center  of  the  circle,  instead  of  the  bisectors  of 
the  angles.  The  given  situation  has  been  connected  with  the  wrong  response, 

A comparison  of  the  methods  of  construction  for  inscribed  and  circumscribed 
circles  and  practice  in  constructing  both  will  aid  in  forming  the  correct  bonds, 

8,  Vague  explanation  of  construction.  Such  an  error  may  occur  in 
the  explanation  of  a construction  line  required  for  the  proof  of  a theorem  or 
in  a construction  problem.  For  example,  diameter  that  bisects  the  chord  is 
perpendicular  to  the  chord  and  bisects  the  arc  subtended  by  it,”  The  plan  of 
proof  of  this  theorem  is  dependent  on  a clear  understanding  of  the  facts  given. 
The  explanation,  "draw  a diameter”  may  lead  to  the  false  assiLmption  that  this 
diameter  is  perpendicular  to  the  chord,  ’’Draw  a diameter  through  the  mid  point 
of  the  chord,”  aids  in  setting  forth  correctly  the  given  facts  in  contrast  to 
those  requiring  proof.  Clear,  definite  explanation  facilitates  proof  in  a 
construction  problem.  The  bonds  between  a construction  and  concise,  exact 
explanations  should  be  formed  and  maintained  by  use.  Dissatisfaction  with  vague 
explanation  will  tend  to  weaken  the  uiidesirable  bonds, 

9.  Inaccurate,  Under  this  head  are  classed  freehand  drawings  or 
approxiTiate  sketches  of  figures  whose  construction  is  required.  For  example,  in 
response  to  '’From  an  outside  point  construct  a tangent  to  a circle,”  a pupil 
using  a straight  edge, by  trial  and  error  process,  draws  a line  which  passes 
through  the  point  and  appears  to  touch  the  circle  at  only  one  point.  The  wrong 
response  has  been  joined  to  the  stimulus  ’’construct”.  This  undesirable  associa- 
tion is  due  to  two  factors.  First,  an  approximate  sketch  in  response  to  the 
stimulus  ’’construct”  has  been  accepted  as  satisfactory.  Perhaps  no  definite 


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40 

distinction  has  been  made  between  "sketch”  and  "construct".  And  second,  the 
familiar  element  of  the  situation  - tangent  - and  the  frequent  response  - "A 
line  that  touches  the  circle  at  only  one  point"-  are  prepotent.  Special  pre- 
paration is  necessary  to  overcome  this  association  and  to  establish  the  con- 
nection between  the  situation  "construct  a tangent"  and  the  response  "a  perpen- 
dicular to  a radius  at  its  outer  extremity  is  tangent  to  the  circle." 

10.  Special  case.  A method  of  construction  that  gives  a satisfactory 
result  under  special  conditions,  but  which  is  not  generally  applicable  to  the 
situation,  is  used.  For  example,  a pupil  constructs  "on  a given  base  a triangle 
equal  to  a given  parallelogram"  by  using  the  base  of  the  parallelogram  for  the 
base  of  the  triangle  and  doubling  the  altitude.  Doubling  the  altitude  would 
fall  to  give  the  required  triangle  if  any  other  base  was  used.  Sometimes  this 
is  undoubtedly  a bluff.  The  pupil  does  not  know  how  to  construct  the  figure 
under  any  other  conditions  but  hopes  to  "get  by"  by  satisfying  the  letter  rather 
than  the  spirit  of  the  requirement.  In  other  cases  the  pupil  is  not  aware  that 
he  is  limiting  the  conditions  of  the  problem.  In  either  case  the  response 
should  be  discouraged.  Calling  attention  to  the  respect  in  vdiich  the  response 
is  unsatisfactory  and  giving  no  credit  for  it  will  encourage  the  pupil  to  watch 
for  and  avoid  special  cases. 

11.  Complete  inability  to  construct.  The  pupil  considers  the  ques- 
he 

tlon.  After^ carefully  numbers  and  copies  the  question,  he  looks  at  it  for  some 
time,  hoping  for  an  inspiration.  He  finds  nothing  in  his  past  experience  sug- 
gestive of  a solution.  Be  records  no  attempt  at  a solution.  He  gives  up  be- 
cause he  does  not  know  how  to  begin.  Either  the  question  is  not  suitable  or  his 
past  experience  is  at  fault,  A situation  requiring  the  use  of  new  elements  or 
new  combinations  of  known  elements  is  not  suited  to  the  ability  of  any  except  the 
upper  third  of  pupils.  Past  experience  may  be  unsatisfactory  from  two  points 


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V"'  ; '■■ ' ■ 

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f ‘ hP>aJ5.\»  Ofi  otf’iOfO'j  wll  .3)J?Jiicas  f,  1&  QV.!^i08  . 

cl  c.  ■ ’^  i , w , • *i‘.*>i  .-'I"  hm  woocii  ^Oc  4|iOX'  ,'| 

m ' • ■ 

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of  view.  The  bonds  involved  In  needed  past  erperlence  may  be  too  weak  to 
facilitate  recall.  These  bonds  may  have  been  formed  in  some  way  other  than 
that  in  vhich  they  are  now  required  to  act.  Additional  experience  in  genuine 
situations  will  Increase  the  pupil's  ability. 

12.  Omission  of  statement  of  construction.  A definite  statement  that 

! 

"IB  is  a line  through  a given  point.  A, tangent  to  the  given  circle  at  point  | 
B,"  makes  the  work  clearer  to  another  who  inspects  it.  The  pupil's  comparison  | 
of  such  a statement  with  a requirement  of  the  problem  tells  him  that  he  has  or  i 
has  not  fulfilled  the  requirements.  The  statement  of  the  construction  also 
sets  forth  definitely  the  fact  to  be  proved.  It  should  be  required  for  every 
construction. 

13.  Unnecessary  construction.  A small  penalty  should  be  attached  to 
adding  an  unnecessary  line  to  a figure.  It  tends  to  confuse  a pupil. 

The  nature  of  the  error  in  proof. 

1.  Wrong  authority.  For  example,  two  sides  of  a triangle  are 
equal  because  "A  bisector  divides  into  two  equal  parts."  The  pupil  evidently 
feels  that  the  bisector  divides  the  triangle  into  two  equal  parts.  The  line 
is  the  bisector  of  the  vertex  angle  of  the  triangle.  It  divides  the  triangle 
into  two  congruent  triangles  only  in  the  case  of  an  Isoscles  triangle.  It  is 
possible  that  the  pupil  is  thinking  of  that  case  but  falls  to  realize  the  limi- 
tation of  its  application.  It  is  more  probable  that  the  difficulty  is  due  to  a 
confusion  resulting  from  the  similarity  of  the  words  "triangle"  and  "angle." 
Considerable  practice  with  illustrative  material  is  sometimes  needed  before  a 
pupil  gets  clearly  in  mind  the  two  meanings.  Sven  after  he  understands  the  die-! 
tlnctlon  a tendency  to  use  the  wrong  name  may  persist.  The  Indiscriminate  use 
of  the  words  results  in  a confusion  in  the  meaning  of  the  early  theorem  of  angle 
and  triangle.  His  abbreviated  statement  of  the  theorem  favors  the  persistence 
of  the  difficulty. 


■ .Ii-.;:.  , ■ i :.l  i.  vioviJ  vVJiCci  c*.'.  ,T‘ii.iv 

i-.;  •■  jV  1 • 1 1 Vti'i 

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'A  • ■ ■•-  . " li;  ‘i'l  ‘ ' -..illMsfi  fon  cnti 

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; iJv,  : -.  j.^Qi-Alc,  iH<1:  JtciJ  r.Iot'. 

■>;■'■  - .:  • uiA;,''- *,o  ; d* 'u;* ' v- ■ J •.■. 'rv'C.  iiA  ai 

. -A  C'  :ri  OA’? 

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A.-  * " '2'  11  . :1a I : is  *?I  ■'•. l>  noiSfW 

> 

.’  ! .‘  ■ ’ . ll  r< ; i - '"Max*.':-  'Ici«v  • -.J  iUOv 

.■y:  . ...  ■:,  ;•'■  • I tfJ  I r.i i ' X tqjJC 

. ' 3.TKA’  iw'.'f  ' ; ■ a ;■•.■.■ 

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42 

The  same  authority  Is  used  to  prove  two  alternate  interior  angles 
equal.  Here  the  fact  used  has  little  meaning  to  the  pupil.  It  Is  one  of  j 

several  facts  which  may  be  used  to  prove  two  angles  equal.  It  probably  seems  I 

to  the  pupil  as  good  a guess  as  any.  This  case  is  typical  of  a large  group  of  j 

errors  made  by  a class  of  pupils  found  in  every  school  - a class  very  slow  to  j 

grasp  the  meaning  of  and  ability  to  use  a logical  proof.  Intuitional  or  con-  S 

I 

struct ive  geometry  naay  be  interesting  to  them  but  offers  no  aid.  Unless  | 

special  supplementary  experience  with  simple  demonstrations  and  a slow  rate  of 
advance  can  be  arranged,  these  pupils  can  succeed  in  a class  of  varied  abllitleE 
only  by  repeating  the  course. 

2.  Absolute  inability.  Absolute  inability  is  frequent  among  the  grouj 
of  pupils  mentioned  above.  Careful  preparation  for  the  bonds  to  be  formed,  and 
a large  amount  of  practice  with  simple  situations  is  needed.  This  will  necessi- 
tate slow  progress  at  the  start.  Either  the  time  devoted  to  demonstrative 
geometry  must  be  increased  or  the  number  of  theorems  decreased.  The  mental 
ability  of  the  pupil  and  his  probable  future  vocation  must  be  considered  in 
determining  which  is  preferable  in  each  case.  Absolute  inability  to  prove  an 
exercise  Involves  the  problems  suggested  for  inability  to  construct. 

3.  Omission  of  authority.  Authority  for  the  last  step  of  a proof  or  | 
for  a step  following  directly  from  the  previous  step  is  often  omitted.  The  | 
practice  of  requiring  no  authority  for  a step  following  directly  from  the  pre- 
vious one  is  confusing  to  the  pupil.  When  is  a reason  necessary  and  when  not? 

A parallel  column  arrangement  of  steps  and  authorities  facilitates  the  develop- 
ment of  the  habit  of  giving  a reason  for  each  step.  A gap  in  either  column 
suggests  a search  for  a suitable  fact.  I^hilure  to  find  the  right  fact  is  then 
the  only  explanation  for  an  omission.  Failure  may  result  because  the  stock  of 
facts  is  too  limited  or  because  they  are  not  stored  in  usable  form. 


' , lUi-  ' ! ' - ■’  .'V  w L ' 

T:  Xvi- 

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f 

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43 

4.  Omission  of  steps.  What  has  been  said  of  omission  of  authority  ap- 

plies to  omission  of  steps  also.  1 step  is  often  omitted,  however,  when  the 
pupil  is  not  aware  of  the  omission.  i?or  example,  a pupil  proves  two  triangles 
congruent  by  two  angles  and  an  Included  side  but  shows  only  one  pair  of  equal 
angles,  A group  of  steps  constituting  a distinct  section  of  the  proof  may  be 
omitted.  For  example,  ''In  the  same  or  equal  circles  the  greater  of  two  unequal 
arcs  is  subtended  by  the  greater  chord,”  Given;  arc  CD  is  less  than  arc  AB,  To 
prove;  chord  CD  less  than  chord  AB,  The  pupil  gives  an  Indirect  proof  assuming 
that  if  CD  is  not  less  than  AB  it  is  equal  to  AB.  He  fails  to  consider  the  , 

f 

i 

possibility  that  CD  may  be  greater  than  AB,  In  both  examples  the  attention  is  > 

i 

centered  on  one  element  of  the  proof  to  the  exclusion  of  the  other.  A step  ■ 
necessary  for  a logical  proof  is  sometimes  omitted  because  it  appears  so  obvious  , 
to  the  pupil.  If  his  attention  is  called  to  his  ovm  difficulties  in  explaining  | 
a step  considered  obvious  by  the  author  of  his  text,  the  pupil  readily  compre-  I 
hends  the  additional  clearness  secured  of  writing  down  each  step  of  the  proof. 

”It  is  obvious”  is  sometimes  used  by  a pupil  to  cover  up  vagueness  in  his 
own  mind.  Such  vagueness  can  be  discovered  and  cleared  up  at  once  by  completing 
the  proof, 

1 

5.  Wrong  step.  A pupil  says,  ’’angle  1 equals  angle  2 because  homologous 

! 

angles  of  congruent  triangles  are  equal,”  Neither  angle  1 nor  angle  2 is  in  i 
either  one  of  the  pair  of  congruent  triangles.  Another  makes  ”CQ  is  identical 
to  CQ**  one  step  preparatory  to  proving  two  triangles  congruent,  when  C^  is  a lin^ 
part  of  which  is  in  one  triangle  and  part  in  the  other.  In  each  case  the  pupil 
fails  to  examine  critically  the  step  used.  The  inconsistency  is  very  apparent 
to  him  when  his  attention  is  called  to  it, 

6.  Wrong  recall  of  plan  of  proof.  Two  types  of  error  are  grouped  to- 
gether here,  (1)  a plan  suitable  to  one  theorem  is  used  for  the  proof  of  another 
to  which  it  is  not  suited.  A pupil  tries  by  the  indirect  method  to  prove  the 


: :'-i;  ;fo  iijr.  ' .'-f*  ci?**-? 

• ''T-.r  c,T'  . 1 1 C.'  J io:lo  !.?rcr:q 

■ .'  i 

. ..  ■'  _uul  tnei 


^ >*  * ' * ' \ G ; 

' I 


r 

( 

h 


t 


. ’ vf  -ru'  -y:f: -<jZ-  '^Jlliei/saoa 

..i  : ' • 5r  i \o  ::j-' ; i ■ PCLO  .ro 


i;  ;•  ■ i bJ  e to*'  & 

/ 

. * - *:  • .i  '<  ■ \l  .n-£jo  , 

'•:-  •:  * ' -’r  :;  :fvcji  fc-3'i Me«Oi>  • 

•■*:'.•  ■■>:■•  : ■ . : lpp.f..i1h  ...K.  «iV-;  \\ 

■.  . .,.  ♦ ...  r.  i:’*  • ' 

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u 

base  angles  of  an  isoscles  triangle  are  equal.  The  bond  between  theorem  and 
plan  is  not  strong  enough  to  function.  The  bond  between  some  element  of  the 
situation  and  a response  may  be  prepotent.  The  connections  between  the  sltuatioa 
”a  theorem  to  prove”  and  the  response  ”what  plan  shall  I follow?”  leads  the 

f 

pupil  to  try  some  known  method.  (2)  A plan  may  be  recalled  up  to  a certain  \ 

j 

point  correctly.  The  pupil  Is  unable  to  determine  the  proper  plan  for  the  re- 
mainder of  the  proof.  For  example,  median  of  a trapezoid  Is  parallel  and 
equal  to  one-half  the  sum  of  the  bases.”  A pupil  makes  the  correct  construction. 
He  proves  the  constructed  figure  a parallelogram,  and  uses  the  auxiliary  triangle 
to  prove  the  median  parallel  to  the  bases,  but  falls  to  recall  how  to  prove  it 
equal  to  one-half  their  sum, 

7 and  13.  Wrong  conclusion  and  hypothesis.  The  most  frequent  error 
la  confusion  of  the  hypothesis  and  conclusion.  For  example, "If  the  diagonals 
of  a quadrilateral  bisect  each  other,  the  figure  Is  a parallelogram.”  Several 
pupils  gave  as  their  conclusion,  the  diagonals  bisect  each  other,  assuming  by 
hypothesis  that  the  figure  was  a parallelogram,  A critical  comparison  of  the 
selected  "given"  and  "to  prove”  with  the  theorem  as  stated  will  correct  these 
errors. 

8.  Failure  to  recall  plan  of  proof.  This  error  Is  similar  to  failure 
to  recall  method  In  a construction  problem, 

9,  Omission  of  proof  of  construction.  Prepotency  of  the  construction  I 
or  inability  to  prove  may  be  the  reason  for  these  omissions, 

10.  Misinterpret  hypothesis.  In  number  13  the  wrong  part  of  the 
statement  is  chosen  as  the  hypothesis.  Here  the  correct  part  of  the  theorem  Is 
considered,  but  an  incorrect  meaning  attached. 

11,  Fhct  wrongly  assumed  by  hypothesis.  "If  the  diagonals  of  a quad- 
rllateral  bisect  each  other,  the  figure  Is  a parallelogram,”  The  hypothesis 


fl'.OTOe’-jlit  .Isjirrfb  9%o-  t^lo^oel  iii  *VOi  fcoiaon  «•*<' 

'■  ' ’■.  . ''  ' ' ;i  * 

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. ■ ...  ...  • ■ ' ' * J 

ij>,  - * . . . -.j^ 

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45 

and  conclusion  are  stated  correctly,  In  symbols,  at  the  beginning  of  the  proof. 
Step  four  says,  **iB  equals  CD,  By  definition  of  a parallelogram."  The  figure 
la  not  a parallelogram  according  to  the  hypothesis.  Critical  examination  of  the 
hypothesis  enables  the  pupil  to  find  and  avoid  the  error, 

12.  Omission  of  part  of  construction.  These  errors  are  similar  to  | 
those  Involving  Incomplete  recall  of  plan  of  proof  In  number  6,  Construction 
lines  and  plan  of  proof  both  In  complete  form  need  to  be  strongly  corjiected  wlthj 
the  theorem. 

13.  See  7 above. 

14.  Use  of  theorem  to  prove  Itself,  Such  an  error  usually  occurs 
In  the  last  stop  of  a theorem.  For  example,  to  prove  "the  angles  opposite  the 
equal  sides  of  an  isoscles  triangle  are  equal."  The  last  step  Is  angle  A equals 
angle  B because  "the  angles  opposite  the  equal  sides  of  an  Isoscles  triangle  are 
equal."  Also  in  proving  the  "median  of  a traoezold  is  parallel  to  the  bases 
and  equal  to  one-half  of  their  sum."  The  laat  step  Is  GP  equals  l/2(AD  plus  BC) 
beoauso"tho  median  of  a trapezoid  Is  equal  to  one-half  of  the  sum  of  the  bases." 
The  symbolism  used  In  the  steps  of  proof  raalres  possible  euch  an  error.  No  one 
would  say  "the  base  angles  of  an  isoscles  triangle  are  equal"  because  "The 

base  angles  of  an  Isoscles  triangle  are  equal."  A translation  of  these  symbols 
baclc  to  words  will  show  the  pupil  the  nature  of  the  error.  The  error  Is  per- 
fectly natural.  The  theorem  Is  fresh  In  his  mind.  The  pupil  realizes  that  he 
has  a situation  which  la  Identical  with  that  described  In  the  hypothesis. 
Therefore  his  theorem  aoplies.  The  following  principle  will  help  to  eliminate 
such  an  error.  Only  facta  which  require  no  proof  (axioms)  and  facts  which  have 
been  proved  may  be  used  as  authorities. 

15.  Unnecessary  step.  A pupil  proves  two  pairs  of  sides  and  two 
pairs  of  angles  equal  In  order  to  prove  two  triangles  congruent.  He  does  not 


'I  , . /;*cvS.'3  .'‘f*?  tw^^/'iOfiOO  hCJi 

; ! ■ I 1 i '•  'l  >*.  ■ r-  ‘ , * ■ . ^ * '*i  <«  ‘ ^ . 0 ''  ( . ..  J . 

. '‘..r.;  i r.’ir:  , ' .LT-ri  ■ i v J w i I v;  - ^ J-'M  ul 

: V-  Lr^*  f'  J'  r U'-  ;X;  .fM  - : n^- 

: **  ■ • ' ^ . ..■*>■-'.■*>.*  ' C n ^ i • ^ 

.i  ■ ■!  ;•  :.  -r..  :':<‘’'.Srr_  ' ' <ikz:ir* 

^ - I.  • ::  • ^ '■.  :.  it,  . '.i  ijr-.O  'OO  jg  .a--; 

• . f . \ i V.  • i 

, • • • • ^ t 

. • 4.  . I t-  «■  . 

. ■ ••'  - : . : • b.  " ” .:■  . ..  : V 1 . Jfi-t  ?.  '\0  «u.J.  Itj 

...i  . .i  - . . . ••  r K.'--  ^*TV  ■_. ■J-.  -fi  /u*  "to  is3M«  I^tpo 

■•  ':■;  ' :1&-.  € Ol-tae 

;•  ; I:.  •■  ; • " ..  .'  * lr.}\ir  ci  'Ai. 

'. .L • • . ;•  :.,  \i  ! ■ • ■ : '•..  ‘s  I . ’v,  • ,i :»•?  o-j  i^upo  ia* 

■ . ■:  --  .1  1 ,7  '.  r t ^ oiii.**' 

f 

...  . ;-v;  : ' zi : t ■)^^.;  r.l  .I'tjdff  ai.^ilocfuT^e  eifT 

• ■ : ■■•  ::i.  7.:  :•  ' . :v-£i/  ..‘.'.c  tic 

' •.. . ■ . ■•,■'•  Je-r:.'.’  ac  ^ .•>  aoinric  9fi"yf 

- . ‘ ■ d.:  ./  v^;‘v:  : . jw  :',t  :ioec 

. ' .-  : •■:  .:■.•  . , . : ■ uu-r.  I ..<13  V ■ . 

..  • ■ : oi  .r  .A.'  '*.»'■/  :i:'.l  : ,i  aJ  t-zJi'v  ncUeij^Ia  £>.  a«-; 

..  J i • • !..  • ' J fal  I'l  f.ni'  J„  J..'1  •.■.■  ..viic  ; , .y•^ 

j ■ ■ - '^  "i  ii.  ; T;  .;•.  ■.■1‘  . ■:  . \lu-  . la?  a^Pi 

. . ' i ''L-if  vat',  c •■•  ; jL;.:  vO  ;i>V.  j'Sv  nattd 

' . 1 't  ' .vrs.'-  n-jj.<r  . ..-.-c.  \;“i.-’-iC9Def:nw  ,.i 

. ...  . ;■  ; ■.'  : t-.  ' ui  iccittf  eijcvi 


46 


consider  that  three  equal  parts  are  sufficient  to  prove  two  triangles  congruent, 

16.  Special  case.  This  error  is  of  the  same  nature  as  a special  case 
In  construction.  The  proof  applies  under  special  conditions  only.  For  example, 
to  prove  **the  common  Internal  tangent  of  two  equal  circles  bisects  their  line 
of  centers,"  he  uses  two  tangent  circles.  The  proof  based  on  this  figure  is 
not  generally  applicable  to  two  equal  sides. 

17.  Step  out  of  order.  A common  mistake  Is,  to  say  that  two  angles 
are  equal  because  they  are  homonogous  angles  of  congruent  triangle,  and  then 
use  the  equal  angles  to  show  that  the  triangles  are  congruent.  The  pupil  has 
not  yet  acquired  a satisfactory  knowledge  of  the  nature  of  a logical  proof. 

18.  Assume  a theorem  not  yet  proved.  A pupil  assumes  "the  area  of  . 
a rectangle  equals  the  product  of  Its  base  by  Its  altitude,"  In  order  to  prove 
"the  areas  of  two  rectangles  have  the  same  ratio  as  the  products  of  their 
bases  and  altitudes."  He  does  not  realize  that  the  former  depends  upon  the 
latter  for  Its  proof. 

19.  Substitution  of  quantity  for  one  not  equal.  A half  of  one  lihe 
Is  substituted  for  a half  of  another  line  regardless  of  the  fact  that  the  two 
lines  are  unequal.  A critical  examination  of  the  situation  would  reveal  the 
inappropriateness  of  such  a step, 

20.  Example  Instead  of  proof.  In  one  case  a pupil  drew  a circle  and 
assigned  niunerical  values  to  the  central  angles  and  arcs  in  order  to  Illustrate 
that  "In  the  same  or  equal  circles,  equal  central  angles  cut  out  equal  arcs," 

For  him  the  meaning  of  the  theorem  was  prepotent  over  the  requirement  of  the 
question. 

21.  Wrong  assumption  in  Indirect  proof.  The  negative  of  the  hypothe- 
sis Is  assumed  Instead  of  the  negative  of  the  conclusion,  A pupil  begins  the 
proof  of  "the  alternate  interior  angles  formed  by  two  parallel  lines  are  equal" 


V . • ? rtifcitff':. 

. 1:j1>  .'f  ..  j ai 

i.-.  ' Ci*f*  -ivorjl 

,;  . ^ . aT  .^ilOO  ^0 

..  •»  - ■ I.l  ^ ior 

-iO  lO  . . i - 

t.--  o:.-..-Z'€-  iJ*Opv  UVif. 

;'V  onjF* 

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> '•'  ■ t'  i 

r .'U’i  yvfj  A ; s- tiv 

l'  ■'■"■'  . , .; 

■;  /.?  *J0  .»*%?**■ 

■’.li  , 

-ii'.  V ' . ,.  ■■ic«i  -J.t  .jtiist  n^9‘&/C 

‘rv’  c*i  ip.’i 

'<:■  -i  • •-•  s*  • r. -t  - ,■' 

.'•  .»  ’:  .i  • / , ••  «! 

jS'C-:-''.;  ■•  .1:  SvaH 

..  ■!  o-C- " '•x-.:0!^f  i 

.-If  : A.r-r-r--’  .o:i  , -■” 

v'  •,  • •!.  ;\,.v  ; -jr;  v."i»fca’  £ea?>ifMia 

. ••  r-i’  ii5l  **i  ■' 

:;.••■  '•.*>:  10  rni/.'--'-  — %9^ 

. ■ . 1 Jftet-i.. 

i ■'  l_  ',.  A.;l.  -••  .»f  ‘ .IS'  •■'  • 

■ .'(ij  '^0  -■*  ?-3  ■•’  f ' V ',  I.-  t-‘-  i ’i  i .i  J 

■y 

^I.'  ’to  '.omq: 


47 


by  assuming  that  the  lines  are  not  parallel.  He  has  connected  the  wrong  response 
to  the  situation,  "How  shall  I begin  an  indirect  proof?" 

The  nature  of  the  errors  in  exercises. 

1.  failure  to  analyze  correctly.  One  may  read  an  exercise  so  that  he 
con5>rehends  the  special  facts  in  their  proper  relation  but  fails  to  connect  the 
given  data  with  a known  theorem  upon  which  the  solution  of  the  exercise  depends. 
One  case  will  illustrate.  The  pupil  is  asked  to  find  the  area  of  the  square 
inscribed  in  or  circumscribed  about  a circle  of  radius  4.  He  draws  the  figure 
and  marks  the  length  of  the  radius  correctly.  He  falls  to  connect  the  Pythagor- 
ean theorem  (which  he  has  proved  in  a previous  question)  with  the  Isoscles  right 
triangle  vdiose  hypotenuse  is  the  required  side  of  the  circumscribed  square.  His 
experience  has  been  lacking  in  applications  of  this  theorem  to  situations  com- 
plicated by  other  lines. 

2.  Wrong  interpretation  of  "required".  This  error  differs  from  (1) 
above  in  that  here  the  pupil  does  not  get  the  correct  meaning  of  the  requirement 
of  the  exercise.  Errors  of  this  kind  are  often  due  to  failure  to  attend  to  what 
is  required.  One  pupil  drew  the  correct  figure  in  the  exercise  cited  above  and, 
ignoring  the  square  entirely,  found  the  area  of  the  circle.  Another  pupil  sub- 
tracts the  circumferences  of  the  two  concentric  circles  and  marks  his  answer 
"difference  in  area."  He  stated  the  requirement  of  IPe  problem  correctly  but 
consciously  computed  the  circumferences.  (He  marked  it  circumference.) 

Other  misinterpretations  are  due  to  wrong  meanings  of  words  Involved.  I 
"A  school  house  is  to  be  located  forty  feet  from  each  of  two  intersecting 
roads."  Distance  from  a point  to  a line  does  not  mean  for  the  pupil  the 
perpendicular  distance  from  the  point  to  the  line  so  he  locates  the  school 
house  forty  feet,  measured  along  an  oblique  line,  from  each  road. 

3.  failure  to  recall  formulae  correctly.  The  needed  formulae  may  be 
entirely  forgotten,  the  wrong  one  of  two  possible  formulae  may  come  to  mind,  or 


. ,i  I.  • ’.H.ii  rJt.  • i 

* iiy  ’ .'ritn.  -'w  ' i:  :■**  ’ ' . 

•,.■,-1:?  w 'u.  vii^  “io  ^vf  :u  1.'^ 

'.  -I.  . . ' '*i •'/*,:  :*  ,i 

i ■ ■ * •• . ■ ij  -.ftj-  j-l.:-fic. ••-  vi--t  sitiS^rtfeK^Oo 

•■■  J 1 . . ■.  • •■  '.'■  . C'O!^  ; . .. .,r  i;  „•  r.'tt  coyj»,; 

■ ■'  I ^■..  •,..  ■-.!  ...  ; .'•1 j f j i Iv  jiCfiO  rWi'- 

■-:  ••£/0,;  :'  i i>c ; ’-'iu.vC'-ii  c<  •.  ■ ..i  bodiio^iil 
,.\i  ' J '-  -J  ^0  •'*  •■•  -.^>1 

■ - r.-i's  « .!•.  : v , ic-  E li  ■ ci^r$r)  en^'i^edt  are 

: - J. -J  . niti  :.i  :oc:\  =■  esi'jifl.ui'iv 

»! 

' • ■■  ' ■ 3 ; ;e-  ; li-  -r  cj  :•  .i  .;««  ii-Daai'i-is.x'/  is 

'•*  • \ 

. I ♦ 

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|i 

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■ '•■-  . ' .'  «**■  ••• . , tc-'-r--  "?  Vf'-Tv  j c»flO  ‘ ?'B«t  cl  j> 

M 

V 

. " ’ ■■■■'  ■ . ^ . v '/  l| 

. ■ ■ ' ■■  ..>  I -i  c •'■•.  V #rtl  " 

I . 

< • • ■'  '■•  ■ *•  V ■:  .V'-'  • ■•  o'  . '<.i  i 


• • . ’:f 

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-aagflp 


i' 


48 


the  needed  formulae  may  be  stated  incorrectly,  For  example,  ”How  many  sides 
has  a regular  polygon  if  one  of  the  interior  angles  contains  108  degrees?”  re- 
mains unanswered  for  lack  of  the  formulae,  "The  sum  of  the  interior  angles  of 
a polygon  of  n sides  is  (n-2}  straight  angles”,  or  ”The  sides  of  the  quadri- 
lateral ABCD  are  10,  17,  13,  and  20  respectively.  The  diagonal  AC  is  21,  Find 
the  area,”  The  pupil  attempts  to  determine  the  area  of  each  triangle  by  the 

formula  A = a*  b , He  fails  to  consider  A •V'st  s-a)  ( s-b)  ( s-c) , Several 

2 

pupils  state  the  area  of  a circle  = Ellr,  Additional  experience  with  these 
formulae  in  the  situation  in  which  they  will  be  used  is  necessary  for  their 
effective  recall.  If  the  formulae  are  not  sufficiently  important  to  be  re- 
called, such  exercises  should  be  omitted  or  the  required  formulae  given  as  part 
of  the  data, 

4.  Misinterpretation  of  the  problem,  ”A  tree  is  to  be  planted  10 
feet  from  the  front  wall  and  15  feet  from  the  corner  of  a house.  How  many 
locations  are  possible?  Show  by  diagram,”  One  pupil  located  a line  of  trees 
10  feet  in  front  of  the  house,  and  another  group  in  a circle  15  feet  from  a 
corner,  A second  puts  tirees  15  feet  from  a rear  corner  of  the  house.  A third 
measures  10  feet  from  a front  wall  or  fence  bounding  the  lawn.  A fourth  tries 
to  locate  a tree  which  will  be  15  feet  from  each  of  the  two  front  corners  of 
the  house  and  also  10  feet  from  the  front  wall. 

5.  Wrong  interpretation  of  "given”.  The  pupil  states  the  given 
facts  correctly  but  fails  to  use  them  with  due  consideration  for  their  meaning. 

6.  Omission  of  proof  of  exercise.  What  was  said  of  "omission  of 
proof  of  a construction”  above  applies  here  also. 

7.  Omission  of  part  of  an  exercise.  A pupil  often  fails  to  hold 
two  parts  of  an  exercise  in  his  attention.  He  finds  the  ratio  of  the  area 
of  two  similar  polygons  whose  sides  are  3 and  9,  but  fails  to  consider  the 
ratio  of  their  perimeter.  Perhaps  in  his  eagerness  to  find  the  answer  he 


I 


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49 


fails  to  read  the  second  part  of  the  question.  A critical  examination  of  the 
question  and  his  response  would  reveal  to  him  the  omission. 

8.  Complete  inability,  A scanty  supply  of  facts  from  which  to  choose 
or  an  xmorganized  stock  results  in  frequent  inability. 

The  nature  of  errors  in  quotation. 

1.  Failure  to  recall  theorem  correctly.  This  group  includes  theorems 
misquoted  in  a proof  and  also  theorems  called  for  directly.  A slight  change 

in  wording  ai&y  change  the  meaning  entirely.  Fhr  example,  "If  two  triangles 
have  three  angles  (sides  is  correct)  of  one  equal  to  three  angles  of  the  other 
the  triangles  are  congruent."  Or,  "The  areas  of  two  similar  polygons  have  the 
same  ratio  as  (the  squares  of)  their  homologous  sides."  Frequent  experience 
with  the  correct  meaning  will  strengthen  the  desired  bonds.  The  bonds  involved 
in  erroneous  statements  will  weaken  if  continually  greeted  by  dissatisfaction. 

2.  Failure  to  recall  theorem.  These  differ  from  the  errors  in 
(1)  above  in  that  no  attempt  was  made  to  state  the  theorem  required. 

3.  Failure  to  recall  definition  correctly.  As  in  (1)  above,  this 
group  Includes  definitions  stated  Incorrectly  as  authority  in  a proof  as  well 
as  those  required  directly.  A typical  example  is  "If  one  pair  of  opposite 
sides  are  parallel  the  quadrilateral  is  a parallelogram," 

4.  Failure  to  recall  definition.  In  these  cases  no  attempt  was  made 


to  define 


*■ 


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50 


Chapter  III 

CAUSES  OF  MISTAKES  AKU  REMEDIAL  SUG3ES7I0R3 

The  classification  of  errors  in  Table  II  according  to  mental  processes 
suggests  a plan  for  the  consideration  of  causes  of  pupils*  mistakes  and  methods 
of  correcting  or  avoiding  them. 

Among  the  classified  errors,  omissions  are  most  frequent.  They  are  of 
two  kinds,  conscious  and  unconscious.  The  former  indicate  inability  to  respond 
to  some  part  of  a situation.  The  pupil  goes  through  elaborate  preparations  for 
a required  analysis  or  proof  of  an  exercise  whose  solution  he  has  already  intui- 
tively discovered,  setting  down  what  is  given  and  what  is  required,  sometimes 
drawing  a figure  which  is  a work  of  art,  but  proceeds  no  further.  His  careful 
beginnings  are  a vain  attempt  to  find,  by  going  as  far  as  he  can,  some  suggestion 
for  doing  the  part  he  does  not  know  how  to  do. 

He  consciously  omits  the  proof  of  a construction  or  exercise  because 
he  cannot  shift  his  point  of  view  from  the  facts  on  which  the  construction  or 
solution  were  based  to  a very  different  set  of  principles  upon  which  the  proof 
depends.  He  omits  the  authority  for  a step  in  a proof  because  he  fails  to  dis- 
cover in  his  stock  of  stored  up  experiences  any  fact  orprinciple  which  applies 
to  this  new  situation. 

As  previously  suggested,  this  failure  may  be  due  to  a scanty  supply 
or  disorderly  arrangement  of  experiences.  In  any  case  conscious  omission  is 
partial  inability  and  differs  from  complete  inability  in  extent  only.  Addition- 
al experiences  will  help  him  by  increasing  the  number  or  strength  of  the  bonds 
between  the  situation  and  the  response.  Careful  and  elastic  organization  of  past 
experiences  in  as  many  ways  as  possible  will  make  his  stock  more  available. 


V, 


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51 


Association  by  similarity  will  function  freely  If  each  situation  Is  viewed  re- 
peatedly from  every  possible  angle  before  It  Is  stored  away.  For  example,  the 
pupil  who  has  classified,  "In  the  same  or  equal  circles  equal  central  angles 
Intercept  equal  arcs"  under  the  three  headings,  equal  circles,  equal  arcs,  and 
equal  central  angles,  will  be  in  a position  to  recall  it  in  a situation  involving 
any  one  of  the  three. 

In  contrast  to  these  omissions  of  which  the  pupil  Is  fully  aware  are 
the  cases  where  he  utterly  oblivious  of  the  incompleteness  of  his  response.  When 
his  attention  1s  called  to  the  omission  he  may  or  may  not  readily  recognize  it  as 
a necessary  part.  He  decides  instantly,  upon  an  examination  of  his  response, that 
he  was  very  dull  or  very  absent  minded,  to  comoare  the  perimeters  of  two  inscribed 
squares  but  to  fall  to  compare  their  areas,  or  that  it  was  very  careless  of  him 
to  omit  the  definition  of  a word  whose  meaning  he  knows.  The  explanation  of  such 
omissions  Is  concentration  on  some  other  element  of  the  situation,  often  a more 
difficult  one.  Critical  examination  of  the  response  to  see  if  it  meets  the  re- 
quirements of  the  situation  will  eliminate  such  omissions. 

Other  unconscious  omissions  are  not  readily  recognized  as  omissions  of 
vital  elements.  The  pupil  thinks  it  foolish  to  write  down  as  a separate  step 
that  "two  radii  are  equal",  a fact  essential  to  the  progressive  arrangement  of 
the  proof,  but  perfectly  obvious  to  him;  or  he  sees  no  sense  in  stating  what  he 
has  constructed,  after  telling  what  v;as  required  and  how  he  did  it. 

Two  factors  contribute  to  this  attitude.  First,  the  pupil  has  a one- 
sided view  of  the  situation.  Because  the  Incomplete  form  conveys  the  desired 
meaning  to  him,  he  can  see  no  reason  for  completeness.  He  does  not  realize  that 
a formal  demonstration  should  present  In  full  all  the  facts  and  relations  neces- 
sary to  the  conclusion.  He  fails  to  consider  that  the  readers  may  not  be  able  to 
Insert  between  the  steps  of  the  proof  those  facts  and  relations  which  are^erfectlj 
obvious  to  the  vTlter.  He  expects  the  teacher  to  credit  him  with  a complete 


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52 


proof  although  he  gives  no  evidence  either  that  he  recognizes  its  abbreviated 
form  or  that  he  taiows  what  steps  are  implied.  Personal  experience  with  trying 
to  fill  in  the  steps  slurred  over  by  the  ’’hence  it  is  obvious”  phrase,  common 
in  the  older  text  books,  is  very  effective  in  changing  his  point  of  view. 

The  outlines  of  proofs  given  in  many  of  the  modern  texts,  merely  sug- 
gesting the  general  plan  of  procedure  and  leaving  the  details  to  be  filled  in 
by  the  pupil,  suggest  to  him  such  an  abridged  form  for  presenting  a demonstration 
Special  instruction  is  required  to  offset  the  force  of  this  suggestion.  The  mean 
ings  ’’outline  of  proof”  and  ’’demonstration”  must  be  clearly  distinguished. 

Outlines  of  proof  often  are  very  valuable.  An  outline  of  the  geometric 
proof  of  the  Pythagorean  theorem  facilitates  recall  of  the  plan  of  proof  and 
enables  the  pupil  at  any  time  to  work  out  the  details  for  a complete  demonstra- 
tion. All  the  early  theorems  should  be  proved  completely.  Exercises  involving 
the  use  of  congruent  triangles  offer  a good  opportunity  to  develop,  in  definite 
and  comparatively  simple  situations,  the  nature  of  a demonstration.  Outline  of 
proof  may  be  introduced  as  an  aid  to  recall  when  the  pupil  comes  in  contact  with 
more  complex  proofs.  The  emphasis  in  review  work  should  be  put  on  plan  or  out- 
line of  proof  rather  than  on  complete  demonstration. 

Closely  related  to  and  to  a large  extent  responsible  for  this  one-sided 
view  of  the  requirements  of  demonstration  is  the  somewhat  vague  and  lax  practice 
commonly  accepted  as  proof.  If  an  indefinite  statement  of  one  or  more  of  the 
facts  fundamental  to  a proof  is  accepted  as  a proof,  we  cannot  expect  the  pupil 
to  acquire  the  correct  idea  of  proof  or  to  give  a complete  response  to  a situa- 
tion requiring  proof.  Common  practice  must  constantly  reenforce  the  distinction 
between  outline  and  demonstration.  Requirement  should  at  ell  times  be  very 
specific  for  either  the  one  or  the  other.  Neither  should  be  accepted  as  a 
satisfactory  response  when  the  other  is  called  for. 

Such  directions  as,  ”State  two  theorems  required  in  the  proof  of  this 


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53 


exercise,"  "Give  the  authority  for  the  last  step  in  the  proof  of  this  theorem", 
"Give  complete  proof  of  the  first  case  and  outline  the  proof  for  cases  two  and 
three  of  the  following  theorem",  "Construct  and  prove",  or  "iinalyze  the  con- 
struction and  give  proof",  are  specific  as  to  requirement  and  conducive  to 
satisfactory  responses. 

Next  after  omission  of  frequency  comes  error  involving  memory.  Some 
of  these  mistakes  are  due  to  failure  to  recall,  others  to  faulty  recall.  The 
former  is  a case  of  bonds  too  weak  to  function.  In  the  latter  the  wrong  re- 
sponse has  been  connected  with  a situation. 

The  function  of  memory  in  geometry  and  the  degree  to  which  it  should 
be  developed  has  been  a much  discussed  subject.  Some  writers  on  method  of  teach- 
ing have  decried  memory  and  lauded  reasoning.  No  one  has  listed  improvement  of 
recall  among  the  chief  objectives  of  the  study  of  geometry.  Rote  memory,  ac- 
companied by  a minimum  of  meaning,  has  little  or  no  place.  No  progress,  hov/- 
ever,  is  possible  xonless  the  bonds  that  connect  a stimulus  and  response  are 
developed  to  a strength  sufficient  to  insure  a prcmot  response  whenever  the 
stimulus  occurs,  ouch  habitual  response  to  a total  situation  or  an  element  of 
a situation  is  recall  or  memory.  The  more  the  pupil  understands  the  full  mean- 
ing of  the  situation  and  of  its  elements,  the  more  chance  he  has  of  recognizing 
a' like  situation  in  new  surroundings.  The  study  of  geometry  requires  repetition 
of  a response  in  the  same  identical  situation  in  some  cases,  A pupil  should 
habituate  the  connection  between  the  theorem  "If  two  lines  are  cut  by  a trans- 
versal so  that  the  alternate  interior  angles  are  equal  the  lines  are  parallel", 
and  the  given  proof  because  it  is  a useful  model  of  an  indirect  proof.  The 
connection  betw'een  a theorem,  the  requisite  construction,  and  plan  of  proof 
oug^t  to  become  habitual. 

Novel  geometrical  situations  demand  the  recognition  of  similar  elements 
joined  to  different  concomitants  and  the  selection  of  one  of  the  responses  with 


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54 


which  that  element  has  been  connected.  In  either  case  no  response  is  possible 
unless  the  bonds  connecting  stimulus  and  response  are  ready  to  act.  Reasoning 
or  purposive  thinking  is  impossible  unless  the  pupil  has  a definite  response  at- 
tached to  those  factors  of  a new  situation  which  he  has  met  in  past  experience. 
Habitual  response  or  recall,  then,  is  the  foundation  on  which  reasoning  is  built. 
There  is  no  antagonism  between  the  two, 

A study  of  our  data  suggests  that  additional  practice  is  desirable 
especially  to  further  (1)  correct  quotation  of  theorems,  (2)  ready  recall  of 
theorems,  and  (3)  connection  between  a theorem  and  correct  plan  of  proof  and 
construction  lines.  In  each  case  memorization  of  the  idea  is  preferable  to  memor- 
ization of  bare  words.  We  must  not  forget,  however,  that  meanings  develop  with 
use  and  full  meaning  can  come  only  as  a result  of  a maximum  of  varied  experience. 

The  third  group  of  mistakes  are  those  related  to  analysis.  Thorndike 
suggests^  that  ''All  learning  is  analytic.  (1)  The  bond  formed  never  leads  from 
absolutely  the  entire  situation  or  state  of  affairs  at  the  moment.  (2)  Within 
any  bond  formed  there  are  always  minor  bonds  from  parts  of  the  situation  to  parts 
of  the  response,  each  of  which  has  a certain  degree  of  independence,  so  that  if 
that  part  of  the  situation  occurs  in  a new  context,  that  part  of  the  response  has 
a certain  tendency  to  appear  without  its  old  accompaniments." 

When  such  a fact  appears  in  a new  context  it  tends  to  provoke  the  total 
was 

response  that  bound  to  it,  or  tends  especially  to  provoke  the  minor  features  of 
that  total  response  which  was  especially  bound  to  it.  All  behavior  is  selective, 
but  certain  features  of  it  are  emphatically  so,  "In  meeting  novel  problems  the 
mental  set  is  likely  to  be  one  which  rejects  one  after  another  response  as  their 
unfitness  to  satisfy  a certain  desideratum  appears."  Separation  of  a subtle 
element  from  the  total  situation  in  which  it  inheres  and  the  acquisition  of  some 
constant  element  of  response  to  it  is  typical  of  analysis. 

1,  Educational  Psychology,  Briefer  Course,  p.l53. 

2.  Thorndike,  "Educational  Psychology",  p. 158. 


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55 


We  have  noted  above,  ("Psychology  of  irithmetlc"  see  page  14  ) the  three 
means  which  Thorndike  suggests  to  facilitate  analysis:  (1)  piecemeal  examination 
of  situation,  (2)  response  to  many  situations  each  containing  the  common  element, 
and  (3)  contrast  of  the  element  to  an  opposite  or  imlike  element.  We  have  sot 
a situation  before  a pupil  and  asked  him  to  analyze  it.  Our  results  will  be  more 
satisfactory  if  we  arrange  the  pupil’s  experience  formally  or  informally  to  pre- 
pare him  to  abstract  the  desired  element  and  attack  a constant  response  to  it. 

We  attempt  this  consciously  now  for  a few  elements  by  repeated  application  of 
certain  principles  to  varied  situations.  For  example,  "To  prove  four  line  seg- 
ments proportional,  try  to  prove  them  homologous  sides  of  similar  triangles." 

Absolute  inability  ranks  next  in  order  of  frequency.  It  may  be  due 
to  failure  to  recall  or  failure  to  analyze  and  is  like  partial  inability  except 
more  extensive.  The  emotional  excitement  of  an  examination  may  produce  a mental 
set  of  hopelessness  or  of  discouragement.  Under  this  influence  a pupil  may 
give  up  a question  with  which  under  other  conditions  he  may  be  successful  at 
least  in  part. 

Illogical  errors  form  a very  small  percent  of  the  total  errors.  The 
pupil  more  often  does  not  know  what  to  put  in  a proof  than  how  to  put  it  together. 

Throughout  our  discussion  of  pupils’  mistakes  certain  difficulties 
constantly  reappear.  (1).  Bonds  have  been  developed  to  insufficient  strength. 

(2).  The  pupil  does  not  pick  out  the  known  elements  of  a situation.  (3)  The 
allkeness  of  two  situations  is  not  recognized.  (4).  He  has  had  too  little  train- 
ing in  abstraction.  (5).  Too  little  preparation  is  made  for  meeting  novel 
situations. 

Several  factors  contribute  to  this  situation: 

1.  The  teacher  does  not  have  clearly  in  mind  the  objectives  in  teaching 
geometry.  (They  are  in  dispute  and  constantly  changing). 

2.  He  is  not  aware  of  the  different  processes  Involved  in  the  study  of 

geometry. 


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56 


3.  His  Icnowledge  of  the  psychology  of  learning  in  general  and  the  psych- 
ology of  geometry  in  particular  is  insufficient  to  the  situation.  As  a result  he 
tells  the  pupil  what  to  do  but  fails  to  prepare  him  so  that  he  is  able  to  do  it, 

4,  He  divides  his  effort  over  too  large  a field  to  be  effective. 

The  following  guiding  principles  are  offered  as  remedial  suggestions; 

1,  Present  each  fact,  idea,  and  principle  with  as  full  meaning  as 

possible. 

2,  Develop  further  meanings  through  use  in  varied  situations, 

3,  For  bonds  which  will  be  required  later,  give  sufficient  practice 
to  insure  readiness  to  act  (especially  bonds  betvreen  theorem,  construction,  and 
plan  of  proof), 

4,  Progress  slowly  at  the  beginning.  Provide  supplementary  help  for 
weak  students, 

5,  To  facilitate  recall  organize  facts  and  principles  in  as  many  ways 
as  possible, 

6,  Provide  specific  practice  in  transition  from  the  field  of  space  per- 
ception to  logical  demonstration  and  vice  versa, 

7,  Follow  Thorndike’ s suggestions,  noted  above,  for  improving  analysis, 

8,  State  requirements  definitely.  Hold  pupil  strictly  to  requirement, 
thus  use  dissatisfaction  to  weaken  the  connection  between  wrong  response  and 
situation, 

9,  Use  parallel  column  arrangement  of  steps  and  authorities  in  formal 
demonstration.  It  encourages  definite  and  complete  proofs, 

10,  Encourage  critical  examination  of  all  results. 

11,  If  a pupil  fails  to  solve  an  original  problem, give  him  the  necessarji 
help.  Avoid  connecting  the  response  ”I  can’t”  with  a novel  situation. 


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57 


APPENDIX 

Examination  Questions 
Geometry  I. 

I.  Construct 

1.  A triangle  having  given  two  angles  and  the  included  side. 

2.  Through  a point  outside,  construct  a line  parallel  to  a given 
line, 

3.  Bisect  a given  arc. 

4.  Inscribe  a circle  in  a given  triangle. 

II.  Any  tv;o. 

1,  The  sum  of  all  the  angles  formed  at  a point  in  a line,  and  on 
the  sane  side  of  a straight  line,  is  equal  to  two  right  angles. 

2,  From  any  point  D on  the  base  of  the  isoscles  triangle  ABC,DS  and 
DP  are  drawn  parallel  to  the  equal  sides  BC  and  AD  respectively.  Prove  that  the 
perimeter  DEBP  is  equal  to  AB  plus  BG. 

3.  The  exterior  angle  of  a triangle  is  equal  to  the  sum  of  two 
opposite  interior  angles. 

4.  A diagonal  of  a parallelogram  divides  it  into  two  triangles 
equal  in  all  respects. 


III.  Prove:  If  two  strai^t  lines  are  cut  by  a' transversal  maxing  the 


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Any  three  questions 


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V.  1.  state  ten  cases  when  two  angles  are  equal, 

2,  State  five  cases  when  lines  are  parallel, 

3.  Define:  x^arallelogram,  rectangle,  t ngent,  circle,  radius, 

YI,  Prove:  Parallel  lines  intercept  equal  arcs  on  a circumference. (one 
case. ) 

YII.  Prove;  A diameter  that  "bisects  the  chord  is  perpendicular  to  the 
chord  and  "bisects  the  arc  su"btended  by  it. 

YIII.  Finish  statement  and  prove:  The  sura  of  the  interior  angles  of  a 
triangle  is  equal  to  


quadr i la  t e ral ? 

4,  In  the  same  circle  or  equal  circles,  equal  arcs  on  a circumference 


are  intercepted  "by..... 

X.  At  a given  point  in  a line  only  one  perpendicular  can  "be  erected 


IX,  1. State  three  cases  when  two  triangles  are  equal. 


2,  State  three  cases  v/hen  two  right  angles  are  equal. 


3.  How  many  right  angles  in  the  sum  of  the  interior  angles  of  a 


throuch  the  line 


I'.i  • 4- 

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59 


Geometry  I. 

I.  Construct; 

1*  Through  three  points  not  in  the  same  strai^t  line  construct 
a circumference. 

2.  Inscribe  a circle  in  a given  triangle. 

3.  Consti*uct  a tangent  to  a circle  from  a given  point  outside. 

4.  Construct  a ri^t  angle  triangle  having  given  two  sides  about 
the  right  angle. 

II.  Any  two. 

1.  ABC  is  an  isoscles  triangle,  B3  is  parallel  to  AG.  Prove  that 
BS  bisects  the  exterior  angle  CBD. 

2.  Prove  that  the  exterior  ang^e  of  a triangle  equals  the  sum  of 
two  opposite  interior  angles. 

3.  If  two  angles  at  the  extremities  of  one  base  of  a trapezoid 
are  equal,  the  non-parallel  sides  are  equal. 

4.  The  sxim  of  all  the  angles  formed  about  a point  is  four  right' 

angles . 

III.  Prove:  and  construct  the  figure.  The  line  joining  the  middle  points 


of  tv/o  sides  of  a triangle  is  parallel  to  the  thiri  side, and  equal  to  one-half 
of  it. 

IT. 


I 


Any  three. 

7.  1.  State  eight  cases  when  tv;o  lines  are  parallel, 

2.  Give  the  nuinher  of  right  angles  in  the  sum  of  the  interior  angles 
of  a hexagon. 

3,  State  four  cases  v/hen  right  angle  triangles  are  equal. 

VI.  State  all  the  cases  when  two  triangles  are  equal  and  prove  one. 

7II,  Construct  the  figure  and  prove;  A diameter  perpendicular  to 
a chord  "bisects  the  chord  and  also  the  arc  subtended  by  it. 

7III.  Prove:  The  diagonals  of  a parallelogram  bisect  each  other, 

IX.  Prove:  If  two  circumferences  intersect  the  line  joining  their 
centers  bisects  at  right  angles  their  common  chord, 

X,  1.  Angles  inscribed  in  a semicircle  are 

2.  Squal  arcs  on  a circumference  are  intersected  by  

3.  In  the  same  circle  or  equal  circles,  chords  equally  distant  from 
the  center  are  equal, 

4.  Circumscribe  a square  about  a circle. 


61 


Any  ton. 


Geometry  I. 


I,  Name  five  fundamental  construction  problems  in  Book  I. 

II,  State  the  converse.  If  tv;o  triangles  have  two  angles  of  one  equal 

respectively  to  two  angles  of  the  other  and  the  included  side  unequal 

III,  Prove:  If  two  straight  lines  bisect  each  other  and  their  extremities 
are  joined,  how  many  pairs  of  equal  triangles  are  formed? 

17,  Construct  the  figure  and  state  the  hypothesis  and  conclusion. 

In  the  given  triangle  ABC  the  perpendiculars  from  the  extremities  of  the  base  to 
the  sides  of  the  triangle  are  equal,  prove  the  angles  which  they  make  v;ith  the 
base  are  eqxml  and  the  triangle  is  isoscles, 

7.  Prove:  The  diagonals  of  a rectangle  are  equal, 

71,  In  triangle  ABC,  angle  A equals  60  degrees,  angle  B equals  70 
degrees,  angle  C equals  50  degrees,  '.Thich  is  the  longest  side  of  the  triangle? 

The  shortest?  Give  proof, 

7II.  Is  this  a correct  statement?  "When  two  lines  are  cut  by  a trans- 
versal the  alternate  interior  angles  are  equal," 

7III,  If  the  bisector  of  the  exterior  angle  of  a triangle  is  parallel 
to  the  base,  prove  the  triangle  is  isoscles, 

IX,  If  AB  is  xjarallel  to  CD  and 
SP  is  parallel  to  GH,  prove  angle  1 equals 
angle  3 and  angle  3 plus  angle  16  equal 
two  ri^t  angles, 

f 

X,  State  five  ways  to  prove  that  a quadrilateral  is  a parallelogram, 

XI,  Given  the  perimeter,  to  construct  an  equilateral  triangle, 

XII,  Given  one  side,  to  construct  a square. 


i - 


J; 


r.  * 


"^  ' ' ’ 1 


62 


Geometry  I. 

I.  Define;  Parallel  linos,  triangle,  polygon,  trapezoid,  parallelogram, 
chord,  tangent,  perpendicular,  "bisect,  congraent. 

II.  a.  If  two  straight  lines  are  cut  by  a transversal  making  the  con- 
secutive exterior  angles  supplementary,  the  lines  are  parallel.  State  the  con- 
verse. 

"b.  If  two  parallel  lines  are  cut  by  a transversal,  the  alternate 
exterior  angles  are  eqiial.  State  the  converse. 

III.  Prove;  The  common  external  tangent  of  two  equal  circles  bisects 
the  line  of  centers. 

17.  Two  triangles  are  congruent  if  three  sides  of  one  are  equal  to  three 
sides  of  the  other. 

7.  How  many  degrees  are  there  in  the  interior  angles  of  a regular 
polygon  of  seven  sides? 

71.  Prove;  If  thie  consecutive  angles  of  a quadrilateral  are  supplement- 
ary, the  figure  is  a parallelogram. 

7II.  Prove;  In  the  same  circle  or  eqtial  circles,  chords  equally  distant 
from  the  center  are  equal. 

7III.  A line  throu^  the  center  of  circle  perpendicular  to  the  chord, 
bisects  the  chord  and  its  subtended  arc. 

IX.  Prove;  The  sum  of  the  angles  of  a triangle  equals  180  degrees. 


63 


Geometry  I. 

I.  An  inscribed  angle  is  measured  by  l/s  of  its  intercepted  arc. 

II.  AB  and  CD  are  two  equal  and  intersecting  ciiords.  Prove  triangles 
ABC  and  BCD  congruent . 

III.  If  two  circles  are  tangent  to  each  other  externally  at  point  A,  | 
the  comnon  tangent  which  passes  through  A bisects  the  other  two  common  tangents,  i 

IV.  Construct  a circle  throu^  three  given  points,  not  on  the  same 
strai^t  line. 

V.  Divide  a given  segment  into  5 equal  parts. 

VI.  Hoy;  many  sides  has  a polygon  tlie  sum  of  Y/hose  interior  angles 
exceeds  the  sura  of  the  exterior  angles  by  720  degrees. 

VII.  If  the  diagonals  of  a quadrilateral  bisect  each  other  the  figure 
is  a parallelogram. 

VIII.  If  a line  joins  the  raid-points  of  two  sides  of  a triangle  it  is 
parallel  to  the  thiid  side  and  equal  to  l/2  of  it. 

IX.  If  two  triangles  have  two  sides  of  one  equal  respectively  to  two 
sides  of  the  other,  but  the  included  angle  of  the  first  greater  than  the  in- 
cluded angle  of  the  second,  then  the  third  side  of  the  first  is  greater  than 
the  third  side  of  the  second. 

X.  The  sum  of  the  angles  of  a triangle  is  one  straight  angle. 


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64 


Geom.  I . 

I 

If  DC  - BC 

angle  3 - angle  4 

Prove  triangle  ADC  oongruent- triangle  ABC 

II 

Given  iJD-DC  =CBsrBA 

angle  B • angle  D 
Prove;  angle  ln«angle  2 


III 


Hov;  many  sides  has  a polygonthe  siun  of  whose  interior  angles  is  equal  to 


five  times  the  sum  of  the  exterior  angles? 


17 

The  sum  of  tlie  three  angles  of  any  triangle  is  equal  to  one  strai^t  angle. 
Give  proof. 

7. 


Prove;  If  the  diagonals  of  a quadrilateral  hisect  each  other,  the  figure  is  a 
parallelogram. 

71 

Prove;  The  tangents  to  a circle  from  an  outside  point  are  equal. 

71 1 

(a)  Construct  a circle  which  v/ill  pass  thiu  three  not  in  a line .Ihq)lain  construc- 
tion. 

71 1 

(h)  Construct  a triangle  congruent  toa  given  scalene  triangle. 

71 1 1 

In  the  same  circle  or  in  equal  circles,  if  two  minor  arcs  are  unequal,  then 
their  chords  are  unequal,  the  greater  arc  being  subtended  by  the  greater  chord.. 


I 


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65 


Geometry  II 

I.  Sound  travels  1100  feet  per  second.  If  a cannon  is  fired  from  a certain 

point,  v;hat  is  the  locus  oj^all  persons  who  hear  the  report  at  the  end  of 
3 seconds.  ‘ 

II.  Given  side  b,  angle  B,  and  altitude  to  the  side  c,  required  to  construct 
triangle  ABC. 

III.  Prove:  Two  triangles  are  similar  if  their  horaologus  sides  are  proportional. 

17.  Given  AB  a diameter  of  the  circle  0;  AB  produced  to  0 and  PC  perpendicular 
to  AG;  AB  a line  intersecting  the  circle  in  Q.  Prove  triangle  APG  con- 
gruent to  triangle  AQ3. 

7.  If  the  perimeter  of  a given  field  is  210  rods  and  a side  of  this  field  is 
to  a corresponding  side  of  a similar  field  as  3;2,  find  the  perimeter 
of  the  second  field. 

71.  Gonstract  a square  equal  to  a given  parallelogram.  Explain  and  prove. 

7II.  The  sides  of  a triangle  are  8,  26,  and  30.  Pind  the  radius  of  the  inscribed 
circle . 

7III.  If  a secant  and  a tangent  are  drawn  to  a circle  from  the  same  point  out- 
side the  circle,  the  square  of  the  tangent  is  equal  to  the  product  of 
the  whole  secant  and  its  external  se^nent. 

IX.  The  area  of  a regular  hexagon  inscribed  in  a circle  is  54  \ 3.  V/liat  is  the 
area  of  the  circle. 

X.  Prove:  The  area  of  2 similar  triangles  are  to  each  other  as  the  squares  of 
any  2 homologus  sides. 


66 


&eoraetry  II 

(Choose  ei^t  out  of  ten)* 

1.  Locate  a school  house  40  feet  from  each  of  two  intersect  roads. 

2.  Construct  the  triangle  ABC  having  given,  h,c,h^- 
(Give  analysis,  construction  and  discussion.) 

3.  State  six  conditions  which  inahe  triangles  similar. 

4.  Prove  that  "If  a secant  and  a tan^^nt  are  drawn  to  a circle  from  the  same 
point  outside  the  circle,  the  square  of  the  tangent  is  equal  to  the  product 
of  the  whole  secant  and  its  external  segment." 

5.  Prove  that  "The  area  of  a triangle  equals  one-half  the  product  of  its  base 
and  altitude." 

6.  The  sides  AB,  BC,  CD,  and  DA  of  quadrilateral  ABCD  are  10,  17,  15  and  20 
respectively,  and  the  diagonal  AC  is  21.  Find  the  area  of  the 
quadrilateral . 

7.  Give  the  geometric  proof  of  the  Pythagorean  Theorem. 

Construct  a rectangle  having  given  base  m and  equal  to  two-thirds  a given 
square. 

9.  Prove  that  "The  area  of  a regular  polygon  is  equal  to  one-half  the  product 
of  the  apothera  and  its  perimeter." 

10.  A circular  grass  plot,  100  feet  in  diameter,  is  surrounded  by  a v;allc  4 
feet  wide.  Find  the  area  of  the  v/alk. 


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67 


Geometry  II 

Choose  eight  questions  out  of  nine.  j 

1 (a)  The  poles  of  a telephone  line  are  to  he  each  equidistant  from  two  houses. 
What  is  the  locus  of  tv/o  poles?  Prove  it. 

(b)  A tree  is  planted  10  ft  from  the  front  wall  of  a rectangular  house  and 
15  ft. from  the  comer  of  this  house.  How  many  solutions  are  there? 

Show  by  diagram. 

2,  Prove  the  theorem:  If  tv;o  chords  are  drawn  throu^  a fixed  point  within  a 

circle  the  prod\;ct  of  the  segments  of  one  of  the  chords  is  equal  to 
the  product  of  the  segments  of  the  other. 

3,  Construct  the  mean  proportional  between  tv/o  unequal  segnents.  Prove  it. 

4,  Prove:  The  square  upon  the  hypotenuse  of  a rt. triangle  is  equal  to  the  sum 

of  the  squares  ux/on  the  other  two  legs, 

5,  The  area  of  Polygon  I is  147  sq.  in,  and  its  shortest  side  is  3 in.  V/hat  is 

the  area  of  Polygon  II  whose  shortest  side  is  9 inches?  What  is 
the  ratio  of  their  perimeters? 

6,  Construct  a triangle  equal  to  a given  parallelogram  with  its  base  equal  to 

the  base  of  the  given  parallelogram. 

7,  Find  the  areas  of  the  circumscribed  and  inscribed  squares  of  a circle  whose 

radius  is  4 inches. 

8,  Pind  the  areas  of: 

(a)  a triangle  of  sides  4,6,  and  8 inches, 

(b)  a tra-pezoid  whose  lower  base  is  16  inches  and  v/hose  upper  base  is  one- 
half  the  lov/er  base,  and  v/hose  altitude  is  one-fourth  of  the  s’wUQ 

of  its  two  bases. 

9,  State  five  ways  in  vdiich  triangles  may  be  proved  similar  and  prove  one  of 


them, 


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68 

Geometry  II 

. I.  A circus  ring  is  fifty  feet  in  diameter.  How  raa^iy  times  v/ill  a 

horse  have  to  circle  it  close  to  the  outer  edge  to  ran  one  mile? 

II.  Hov;  can  you  place  a doily  in  the  center  of  a square  table? 

III.  Required  to  divide  a line  segment  into  five  equal  parts,  using 
a sheet  of  ruled  paper. 

IV.  If  a post  six  feet  high  casts  a three  foot  shadow,  how  tall  is  a 
tree  wMch  casts  a thirty  foot  shadow? 

V.  V/liat  is  the  area  of  a triangular  piece  of  ground  when  one  side 
is  forty  feet  and  the  hypotenuse  is  fifty-two  feet? 

VI.  How  much  longer  will  it  tahe  a two  inch  pipe  to  empty  a tank 
than  an  ei^t  inch  one? 

VII.  Ho’w  many  square  feet  of  cement  in  an  equilateral  triangle  six 
feet  on  a side? 

VIII.  How  could  you  make  a six  sided  bird  house  and  cut  the  sides 
so  they  would  fit  and  not  leave  cracks? 

IX.  Hov/  can  I find  the  center  of  a locket  in  order  to  put  a stone  there? 

X.  A man  wished  to  place  eig^t  shrubs  around  his  circular  lily  pond. 

How  did  he  find  the  places  to  set  them? 


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69 


Geometry  II 


I.  Construct  the  perpendicular  "bisectors  of  the  sides  of  an  o"btuse 
triangle,  "here  do  they  meet? 

II.  Divide  a given  line  segment  into  five  equal  parts. 

III.  On  a given  "base  construct  a triangle  equal  to  a given  parallelo- 


i 

I 


f 


gram. 


IV.  If  two  tangents  are  drawn  throu^^  a circle  from  an  oxitside  point, 
the  line  joining  the  point  to  the  center  of  the  circle  bisects  the  arc  "be- 
tween the  tangents. 

V.  The  legs  of  a ri^t  triangle  are  21  and  18.  Find  the  altitude  to 
the  hypotenuse  if  the  segments  of  the  hypotenuse  are  27  and  12. 

VI.  A baseball  diamond  is  a square  whose  side  is  90.  Hov/  far  will 
the  man  on  first  base  have  to  throw  the  ball  to  the  man  on  third? 

VII.  Find  the  area  of  a trapezoid  if  the  altitude  is  seven  and  the 
bases  15  l/3  and  12  l/2, 

VIII.  ’That  is  the  ratio  of  the  perimeter  of  tvvo  squares  inscribed 
in  circles  whose  radii  are  two  and  eight?  Find  the  ratio  of  the  areas  of 
the  squares. 

I 


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70 


BIBLIOGSAPHY 


1.  Garth,  T.H.,  “The  Psychology  of  Riddle  Solution:  An  Experiment  in  Purposive 
Thinking,**  Journal  of  Educational  Psychology,  January  1920,  Vol.  ll,p.  16-S3. 

2.  Judd,  C#H,,  “Psychology  of  High  School  Subjects'*,  Giim  and  Co,,  N.Y.,1915. 

3.  Minnlch,  J,H.,  “An  Investigation  of  Certain  Abilities  Fundamental  to  the 
Study  of  Geometry."  University  of  Pennsylvania,  1918. 

4.  Odell,  C.W,,“A  Study  of  1,000  Errors  in  Latin  Prose  Composition”,  School  and 
Society,  Dec. 31,  1921,  p. 643-646. 

5.  Rogers,  A.L,,  “Teats  of  Mathematical  Ability  and  their  Prognostic  7alue,“ 
Teachers  College,  Columbia  University,  N.Y.,  1918. 

6.  Ruger,  Henry  A.,  “Psychology  of  Efficiency, Study  of  Mechanical  Puzzles."  y 
Archives  of  Psychology,  No. 15,  Vol.2,  Series  2,  June  1910. 

7.  Rugg,  H.O.,  and  Clark,  J.R.,  “Scientific  Method  in  the  Reconstruction  of 
Ninth  Grade  Mathematics,"  University  of  Chicago  Press,  Chicago,  1918, 

8.  Symonds,  P.M.,  “The  Psychology  of  Errors  in  Algebra,"  The  Mathematics  Teacher, 
Feb.  1922,  Vol.15,  p.93. 

9.  Thorndike,  E.L. , “The  Psychology  of  Arithmetic."  Macmillan,  N.Y.,  1922, 

10.  Thorndike,  E.L.,  “Reading  as  Reasoning: A Study  of  Mistakes  in  Paragraph 
Reading."  Journal  of  Educational  Psychology,  June  1917,  Vol.8,p.323. 

11.  Thorndike,  S.L.,  “Understanding  of  Sentences:  A Study  of  Errors  in  Reading." 

Elementary  School  Journal.  October  1917,  Vol.  18,  p.98. 


